research article derivation of field equations in space...

Research ArticleDerivation of Field Equations in Space with the GeometricStructure Generated by Metric and Torsion

Nikolay Yaremenko

Yu A Mitropolsky Department International Mathematical Center JO Mitropolsky Kiev 01601 Ukraine

Correspondence should be addressed to Nikolay Yaremenko mathkievgmailcom

Received 21 August 2014 Revised 29 October 2014 Accepted 29 October 2014 Published 11 December 2014

Academic Editor Kazuharu Bamba

Copyright copy 2014 Nikolay YaremenkoThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is devoted to the derivation of field equations in space with the geometric structure generated by metric and torsiontensors We also study the geometry of the space generated jointly and agreed on by the metric tensor and the torsion tensorWe showed that in such space the structure of the curvature tensor has special features and for this tensor we obtained analogRicci-Jacobi identity and evaluated the gap that occurs at the transition from the original to the image and vice versa in the case ofinfinitely small contoursWe have researched the geodesic lines equationWe introduce the tensor120587

120572120573which is similar to the second

fundamental tensor of hypersurfaces 119884119899minus1 but the structure of this tensor is substantially different from the case of Riemannianspaces with zero torsion Then we obtained formulas which characterize the change of vectors in accompanying basis relative tothis basis itself Taking into considerations our results about the structure of such space we derived from the variation principle thegeneral field equations (electromagnetic and gravitational)

1 Introduction

In this paper we study the properties of the geometry of thespace generated jointly and agreed on by the metric tensorand the torsion tensor so we investigate the spaces withconnection in the presence of the metric tensor We obtainedsome results on the structure of the curvature tensor andconsidered the construction of geodesic lines and an estimateof the gap that occurs when traversing the contour of aparallelogram in these spaces

The investigation of properties of metric spaces and affineconnection spaces began approximately at the beginning ofthe 20th century [1 2] and continued to develop so far [2ndash16]

The importance of this kind of research is on the onehand due to the internal logic of mathematical science basesitself [1 2 9 13] and on the other hand to applications toproblems in physics analytical and theoretical mechanics[3 12] the theory of relativity [7 14ndash16] and continuummechanics and cosmology [10] Fairly well studied Riemannspaces [9] because of the wealth of geometric propertiesand less explored space with affine connection [5] are notsufficiently considered the most interesting geometry which

is obtained by combining geometry and affine connectiongenerated by the metric tensor and this is the subject ofthis work From the theory of spaces with affine connectionit is known that parallel displacement vector depends onpathways that is if the vector is parallel transported at thegiven contour with it return to the starting point we obtainthe other vector than the original (the gap appears) In thespaces which are studied in this work not only do they holda similar statement but also there are new properties of thisgap

The principle of least action (more correctly the principleof stationary action) is the basic variational principle ofparticle and continuum systems Let the starting point be theaction denoted as 119878 of a physical system It is defined as theintegral of the Lagrangian 119871 between two instants of time afunctional of the 119899 generalized coordinates 119902which define theconfiguration of the system

119878 (119902 (119905)) = int

1199052

1199051

119871 (119902 (119905) 119902 (119905)) 119889119905 (1)

where the dot denotes the time derivative and 119905 is timeMath-ematically the principle is 120575119878 = 0 where 120575means a variation

Hindawi Publishing CorporationJournal of GravityVolume 2014 Article ID 420123 13 pageshttpdxdoiorg1011552014420123

2 Journal of Gravity

In applications the statement and definition of action aretaken together

120575int

1199052

1199051

119871 (119902 (119905) 119902 (119905)) 119889119905 = 0 (2)

The action and Lagrangian both contain the dynamics ofthe system for all times The term ldquopathrdquo simply refers to acurve traced out by the system in terms of the coordinates inthe configuration space that is the curve 119902(119905) parameterizedby time On the other hand a Finsler manifold is a differ-entiable manifold together with the structure of an intrinsicquasimetric space in which the length of any rectifiable curve120591 [119886 119887] rarr 119872 is given by the length functional

119878 (120591 (119905)) = int

1199052

1199051

119865 (120591 (119905) 120591 (119905)) 119889119905 (3)

where 119865(119909 sdot) is a Minkowski norm on each tangent space Itis obvious from these definitions that there is a connectionbetween these two concepts which can be realized byHamiltonian formalism Thus any Riemannian space can beregarded as Finsler manifold with the length functional 1198652 =119892119894119895(119909)119889119909

119894

119889119909119895 and so the geodesics of a Finsler manifold are

geodesics of Riemannian spaceThe geodesics of the space that are being studied in

our work (with the geometric structure generated by metric119892119894119895(119909) and torsion 119878119896

119894119895(119909) tensors) are different from geodesics

of corresponding Riemann space (with 119892119894119895(119909)) and so of

geodesics Finsler manifold (with 1198652 = 119892119894119895(119909)119889119909

119894

119889119909119895)

These spaces retain all the properties of geometry ofan affine space but important features associated with thepresence of the metric appear the structure of the curvaturetensor has a specific characteristics as well as an opportunityto assess the gap that occurs at the transportation from theoriginal to the image and conversely in the case of infinitelysmall contours

The main objective of this work-the study of the geomet-ric properties of the space that is generated by metric 119892

119894119895(119909)

and torsion 119878119896

119894119895(119909) tensors obtain the field equations from

variation principle in such spacesWe remind that according to Albert Einstein proposal

the free falling gravitating massive bodies follow geodesic lineIf we postulate this proposal we can obtain some results ofNewton theory as a consequenceWe have another importantassumption of Albert Einstein that the geodesic equation ofmotion can be derived from the field equations for emptyspace

Since we believe that gravitational and electromagneticfields are determined geometric structure of empty space(torsion and curvature) it is interesting to have exampleabout geometric sense of torsion

Now we discuss one example about geometric sense oftorsionWe consider the surface 119878 At point119860 on 119878 construct atangent plane 119875 We choose an arbitrary infinitesimal square119860119861119862119863 in the plane 119875 with vertex 119860 From point 119860 on thesurface 119878 will draw the geodesic in the direction of 119860119861 Wepass along the distance corresponding parameter equal to the

length of119860119861 and get to point 1198611015840 Similarly from119860 on 119878 drawgeodesic towards 119860119863 and get into 1198631015840 We perform a paralleltransportation of vector 119860119863 to point 1198611015840 along the geodesic1198601198611015840 and draw geodesic 1198611015840 along the transportation of this

vector we reach the point 1198621015840 Similarly the vector 119860119861 willmove parallel along the1198601198631015840 and along the transported vectorfrom 119863

1015840 draw a geodesic to get to 11986210158401015840 If torsion is zero then1198621015840

= 11986210158401015840 and geodesic square up to small higher order will

be closed otherwise not In our case due to the presence of themetric can calculate the length of the gap more precisely we canestimate the length of this gap

These considerations are true only up to the second orderrelative to the length of square side If we want more strictresult we must consider the component of curvature tensorNext this example is true only when length of square sidetends to zero that is remains very small in other words ingeneral it is a local property

We go to the discussion of physical interpretation of thisexampleThe physical properties of the space-time (more justspace) are defined by the presence of matter (electromagneticfields and mass) in this space and from the viewpoint ofmathematics are described by the geometrical structure ofspace (torsion andmetric tensors)The empty space (withoutmatter) corresponds to the geometric structure of Euclideanspace (torsion tensor and curvature tensor are identicallyequal to zero) Similarly gravitation (mass and without elec-tromagnetic fields) corresponds to the geometric structure ofRiemannian space (torsion tensor is identically equal to zero)And similarly the electromagnetism (electromagnetic fieldsand without mass) corresponds to the geometric structure ofaffine connection space (curvature tensor is identically equalto zero) In the last two cases the result is conditional (notstrict) because the matter division by the mass and field isconditional

To describe the internal geometry of the subspace issufficient to define two tensors the metric tensor 119892

119894119896and the

torsion tensor 119878119898119894119896 using these tensors one can build connec-

tion of the space and all the geometry of the spaceIf we consider the space of hypersurface 119884119899minus1 as a sub-

space then for a complete description of the geometry itis necessary and sufficient to define the tensors 119886

120572120573and

119879120574

120572120573 If we consider the hypersurface 119884

119899minus1 as a subspacethat embedded in a space 119884

119899 then it is not sufficient toknow tensors 119886

120572120573and 119879

120574

120572120573for identifying the space 119884119899 (for

obtaining the metric tensor 119892119894119896and the torsion tensor 119878119898

119894119896)

to describe the geometry of the ambient space it is necessaryand sufficient to specify exactly how the hypersurface 119884119899minus1 isembedded in it that is specify the embedment it can be donein various ways We assume that Jacobi matrix is known sothe way of embedment hypersurface119884119899minus1 in119884119899 is determined(also the space can be recovered by determination of normalfor the hypersurface 119884119899minus1)

We consider the followingmethod of constructing space-time the space is constructed on the basis of manifoldsby determining these manifolds metric tensor and torsiontensorMetric and torsion tensors are calculated from the dif-ferential equations of the field Hence torsion as the curvature

Journal of Gravity 3

arises from the physical features of the distribution of matterin space-time Roughly the same way as the masses leads tocurvature space-time electromagnetism leads to appearanceof torsion But on the other hand from the mathematicalpoint of view if we assume that the space-time embeddedin Euclidean space of higher dimension then the appearanceof torsion can be explained by violation of the smoothnessembedding Therefore we can conditionally determine thetorsion and curvature by violation of smoothness regardlessof the dimension and embedment

The aim of our paper is to study the property of metricspace with torsion and obtain analog of Einstein-Hilbertequation at such space

This paper is organized as follows In Section 2 somegeneral properties of structure of a metric space with torsionare discussed In Section 3 we present the study of geodesicsin the space with torsion These results can be used inldquogeodesic principlerdquo In Section 4 the theory of hypersurfacesin the space with torsion is dedicated In Section 5 weobtained some interesting relationships which is used inSection 6 In Section 6 we derive analog of Einstein-Hilbert(for electromagnetic and gravitational fields) in case of ametric space with torsion

The main natural assumption that is used below that ascalar product of two any vectors that is transporting parallelalong an arbitrary path does not change of the 20th century[2 8 11ndash13] and continues to develop so far [3ndash7 14ndash22]

2 Structure of a Metric Space with Torsion

There are many ways to represent the physical four-dimensional space where events of our reality are occurringFrom a mathematical point of view there are two possibleconceptions of space geometry which might be identifiedwith the physical space

The first scheme is a generalization of Euclidean geom-etry the geometry of the Riemannian metric that is 119899-dimensional manifold equipped with a field twice covariantsymmetric metric tensor which is non-degenerate 119892

119894119896(119872)

where Det|119892119894119896| = 0 and 119892

119894119896= 119892119896119894 Note that the metric tensor

is chosen arbitrarily but in addition to conditions laid abovewe demand that the manifold was sufficiently smooth

This definition can be rewritten as follows invariantdifferential quadratic form 119892

119894119896119889119909119894

119889119909119896 determined on the

manifold and satisfying the conditions Det|119892119894119896| = 0 119892

119894119896= 119892119896119894

defines the geometry of RiemannAs a consequence of the invariance of the form

1198891199042

= 119892119894119896119889119909119894

119889119909119896

(4)

we find that the coefficients 119892119894119896are forming a tensor field

In this model for the arc length of the curve 119909119894 = 119909119894

(119905)119905 isin [119886 119887] sub 119877 is given by integral

119904 = int

119887

119886

radic119892119894119896119889119909119894119889119909119896

= int

119887

119886

radic119892119894119896(1199091(119905) 119909

119899(119905))

119889119909119894

(119905)

119889119905

119889119909119896

(119905)

119889119905

119889119905

(5)

The second scheme is a generalization of affine geometrythe geometry of affine connection Γ119894

119895119896(119872) that is based on 119899-

dimensional manifoldThe connection Γ

119894

119895119896(119872) is a geometric object on a man-

ifold and is subjected to the law of the transformation fromone coordinate system 119909

119894 to another 1199091198941015840

in the form of

Γ1198941015840

11989510158401198961015840 = Γ119894

119895119896

1205971199091198941015840

120597119909119894

120597119909119895

1205971199091198951015840

120597119909119896

1205971199091198961015840+

1205972

119909119894

1205971199091198951015840

1205971199091198961015840

1205971199091198941015840

120597119909119894 (6)

where the functions Γ119894119895119896are sufficiently smooth

Let along the curve 119909119894 = 119909119894

(119905) 119905 isin [119886 119887] sub 119877 is giventensor field 119860119894 = 119860

119894

(119905) if for each infinitesimal displacementtensor 119889119860119894(119905) coordinates is changing the law

119889119860119894

= minusΓ119894

119895119896119860119895

119889119909119896

(7)

then we say that the tensor 119860119894 is transported parallel to thecurve 119905

Depending on the physical investigated problem one oranother geometricmodel is choosing but as the internal logicand common sense requires that in the physical world thesetwo models coexist together and complement each otherThere is well-known result that in an arbitrary Riemannianspace can always construct a connection Γ

119894

119895119896(119872) An inter-

esting question is the uniqueness of such a construction Ingeneral such a construction Γ119894

119895119896is not unique but completely

natural (in terms of mathematics and physics to a greaterextent) there is the requirement that whenever two vectors119860119894and 119861119894 simultaneous parallel transported along a path (due tothe presence of the connection the transportation is defined)their scalar product does not change (the scalar product isdefined by metric) Mathematically this can be written as thevanishing differential

119889 (119892119894119896119860119894

119861119896

) = 0 (8)

If the requested coefficients Γ119894119895119896

are symmetric namelyΓ119894

119895119896= Γ119894

119896119895 then the connectivity is uniquely defined using a

metricAlways below we would not require the symmetry of

connection And so if the metric 119892119894119896

is defined then ageometric object Γ119894

119895119896subject to certain requirements but still

there is some arbitrariness in the choice of connectedness ofthe space namely we need to define a torsion tensor

119878119894

119895119896equiv Γ119894

119895119896minus Γ119894

119896119895 (9)

then the geometric object Γ119894119895119896that generated the connection

is uniquely determined

Theorem 1 Suppose that a Riemannian space with the metric119892119894119896and in this space is given torsion tensor 119878119894

119895119896-antisymmetric

If demand 119889(119892119894119896119860119894

119861119896

) = 0 for arbitrary 119860119894 and 119861119896 then the

connection (geometric object that defines it) Γ119894119895119896

is uniquelydefined


Proof It is pretty easy to see the truth of this statement itselfbut for further more importantly those symbols and valuesthat relate to the nature of the entered values

We rewrite 119889(119892119894119896119860119894

119861119896

) = 0 as (119892119894119896119897

minus 119892119898119896Γ119898

119894119897minus

119892119894119898Γ119898

119896119897)119860119894

119861119896

119889119909119897

= 0 due to the fact that 119889119860119894 = minusΓ119894

119901119897119860119901

119889119909119897

where Γ119898119894119897 unknown coefficients of connection are a geomet-

ric object and 119889119909119897 are the differentials of coordinates of a

point under infinitesimal displacement along the path 119892119894119896119897

equiv

(120597120597119909119897

)119892119894119896 Since119860119894 119861119896 119889119909119897 are arbitrary the equalities must

be identity relative to 119860119894 119861119896 119889119909119897 By circular permutation weobtain the system of equations

119892119894119896119897

= 119892119898119896Γ119898

119894119897+ 119892119894119898Γ119898

119896119897

119892119897119894119896

= 119892119898119894Γ119898

119897119896+ 119892119897119898Γ119898

119894119896

119892119896119897119894

= 119892119898119897Γ119898

119896119894+ 119892119896119898Γ119898

119897119894

(10)

Since the technique is similar to the classical one then wegive formulas without justification

119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

= 119892119898119896119878119898

119894119897+ 119892119898119897119878119898

119894119896+ 119892119894119898Γ119898

119896119897+ 119892119898119894Γ119898

119897119896 (11)

where 119878119898119894119897= Γ119898

119894119897minus Γ119898

119897119894is torsion tensor and we have

119892119894119898(Γ119898

119896119897+ Γ119898

119897119896)

= 119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

+ 119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894

Γ119901

119896119897+ Γ119901

119897119896

= 119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

+ 119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

(12)

and complementing the obvious equation-definition Γ119901

119896119897minus

Γ119901

119897119896= 119878119901

119896119897 then we obtain

Γ119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

+ 119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

+

1

2

119878119901

119896119897

(13)

Thenwe introduce the notation and from the last formulawe see that

119875119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894) (14)

is geometric object and

119871119901

119896119897equiv

1

2

119878119901

119896119897+

1

2

119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) (15)

is tensorThe geometric object Γ

119901

119896119897 which generate connection

space is completely determined by the tensors 119892119894119896and 119878

119898

119894119896

Therefore the connection Γ119901119896119897is the sum of a geometric object

119875119901

119896119897which is composed of derivatives of the metric tensor 119892

119894119896

and tensor 119871119901119896119897is compiled of 119892

119894119896and the tensor 119878119898

119896119897 namely

Γ119901

119896119897= 119875119901

119896119897+ 119871119901

119896119897 (16)

Remark 2 Tensor 119871119901

119896119897represents the sum of two tensors

symmetric (12)119892119901119894(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) and torsion (12)119878119901

119896119897

Remark 3 It is not difficult to prove the relation

Γ119901

119901119897=

1

2

119892119894119901119897119892119894119901

=

1

radic119892

120597radic119892

120597119909119897

where 119892 = det 1003816100381610038161003816119892119894119896

1003816100381610038161003816 (17)

The next step in building a geometric theory is the con-sideration of the parallel transport tensor-vector 119860119894 which isgiven by

119889119860119894

= minusΓ119894

119901119897119860119901

119889119909119897

(18)

where the coefficients Γ119894119901119897are the connection of space

As further arguments are similar to the classic and oftenrepeated them the presentation of intermediate results willwear schematic character

By a covariant derivative of 119906119894with respect to 119897 we mean

119906119894119897equiv 119906119894119897minus Γ119896

119894119897119906119896

119906119894

119897equiv 119906119894

119897+ Γ119894

119896119897119906119896

(19)

Then we consider the difference

119906119894119897minus 119906119897119894= 119906119894119897minus 119906119897119894minus 119878119896

119894119897119906119896 (20)

During the transition along a parallelogram in the imageto the original polygon the gap 119885

119896 is formed (breaking thecircuit) which can be estimated as follows up to the 2ndorder of smallness relative to sides of a parallelogram

119885119896

= 119878119896

119894119895119860119894

119861119895

1205912

(21)

It is the result of coagulation at the torsion tensor with vectorparties 119860119894120591 and 119861

119895

120591 that express geodetic displacement Thebasic geometric meaning of the torsion tensor is an estimateof the gap (up to 2nd order) at which the open loop shrinks ifthe torsion is zero then the gap will not infinitesimal secondbut higher order smallness

Next we consider the difference of the second orderderivatives

119906119894119897119896

minus 119906119894119896119897

= 119877119901

119896119897119894119906119901+ 119878119902

119896119897119906119894119902 (22)

where we identified

119877119901

119896119897119894equiv Γ119901

119894119896119897minus Γ119901

119894119897119896+ Γ119901

119902119897Γ119902

119894119896minus Γ119901

119902119896Γ119902

119894119897 (23)

119877119901

119896119897119894is curvature tensorSimilarly we have

119906119894

119897119896minus 119906119894

119896119897= minus119877119894

119896119897119901119906119901

+ 119878119902

119896119897119906119894

119902 (24)

Remark 4 Also possible and slightly different way of defini-tion a covariant derivative namely the absolute differential119863119860119894 is first determined using the formula

119863119860119894

equivsim 119889119860119894

+ Γ119894

119895119896119860119895

119889119909119896

= (119860119894

119896+ Γ119894

119895119896119860119895

) 119889119909119896

(25)


Absolute differential can be defined as derivative coeffi-cients all the results obtained with this approach to the con-struction is identical with the analysis which has been madeabove Then more clearly we can assert that for any space topossess absolute parallelism it is a necessary and sufficientcondition that curvature tensor be identity vanishing (recallthat the space is called with absolute parallelism if the resultof the parallel transport of an arbitrary tensor-vector doesnot depend on the choice of path for all points of space) Theproof of this theorem is generally known we note only that itfollows from the formula

119863119863119860119894

minus 119863119863119860119894

= minus119877119894

119896119897119901119860119901 119889119909119896

119889119909119897

(26)

which we could get by folding (24) with 119889119909119896

119889119909119897

All the constructions outlined above are general in naturewithout specifying space further we will investigate thestructure of the tensor 119877119901

119894119896119897 So by definition of (23) we have

119877119901

119894119896119897= Γ119901

119897119894119896minus Γ119901

119897119896119894+ Γ119901

119902119896Γ119902

119897119894minus Γ119901

119902119894Γ119902

119897119896 (27)

Then we use (16) and obtain

119877119901

119894119896119897= 119875119901

119897119894119896+ 119871119901

119897119894119896minus 119875119901

119897119896119894+ 119871119901

119897119896119894+ (119875119901

119902119896+ 119871119901

119902119896) (119875119901

119897119894+ 119871119902

119897119894)

minus (119875119901

119902119894+ 119871119901

119902119894) (119875119901

119897119896+ 119871119902

119897119896)

= 119875119901

119897119894119896minus 119875119901

119897119896119894+ 119875119901

119902119896119875119902

119897119894minus 119875119901

119902119894119875119902

119897119896+ 119871119901

119897119894119896minus 119871119901

119897119896119894

+ 119875119901

119902119896119871119902

119897119894+ 119875119902

119897119894119871119901

119902119896minus 119875119901

119902119894119871119902

119897119896minus 119875119902

119897119896119871119901

119902119894

+ 119871119901

119902119896119871119902

119897119894minus 119871119901

119902119894119871119902

119897119896

(28)

Next we introduce the notation

119875119901

119894119896119897equiv 119875119901

119897119894119896minus 119875119901

119897119896119894+ 119875119901

119902119896119875119902

119897119894+ 119875119901

119902119894119875119902

119897119896(29)

is a tensor like the Riemann curvature tensor composed ofthe metric tensor and its derivatives Consider the following

119885119901

119894119896119897equiv 119871119901

119902119896119871119902

119897119894minus 119871119901

119902119894119871119902

119897119896(30)

is tensor and

119879119901

119894119896119897equiv 119871119901

119897119894119896minus 119871119901

119897119896119894+ 119875119901

119902119896119871119902

119897119894+ 119875119902

119897119894119871119901

119902119896minus 119875119901

119902119894119871119902

119897119896minus 119875119902

119897119896119871119901

119902119894(31)

is tensorIf we take into account that119877119901

119894119896119897is tensor the last assertion

is obvious It is interesting to obtain this important result inanother way namely

119879119901

119894119896119897= 119871119901

119897119894119896minus 119871119901

119897119896119894minus 119871119902

119897119894Γ119901

119902119896+ 119871119901

119902119894Γ119902

119897119896+ 119871119901

119897119902Γ119902

119894119896+ 119871119902

119897119896Γ119901

119902119894

minus 119871119901

119902119896Γ119902

119897119894minus 119871119901

119897119902Γ119902

119896119894+ 119875119901

119902119896119871119902

119897119894+ 119875119902

119897119894119871119902

119902119896

minus 119875119901

119902119894119871119902

119897119896minus 119875119902

119897119896119871119901

119902119894

= 119871119901

119897119894119896minus 119871119901

119897119896119894minus 119871119902

119897119894119871119901

119902119896+ 119871119901

119902119894119871119902

119897119896+ 119871119902

119897119896119871119901

119902119894

minus 119871119901

119902119896119871119902

119897119894+ 119871119901

119897119902119878119902

119894119896

(32)

is obviously the tensor because the absolute derivatives havetensor character

We introduce the notation

119872119901

119894119896119897equiv 119879119901

119894119896119897+ 119885119901

119894119896119897 (33)

then we obtain

119872119901

119894119896119897= 119871119901

119897119894119896minus 119871119901

119897119896119894+ 119871119901

119897119902119878119902

119894119896+ 119871119901

119902119894119871119902

119897119896minus 119871119901

119902119896119871119902

119897119894 (34)

and in the new notation

119877119901

119894119896119897= 119875119901

119894119896119897+119872119901

119894119896119897 (35)

Formula (35) shows that the curvature tensor in generalcases can be represented as the sum of two tensors (suchrepresentation is not accidental it is associatedwith a physicaldescription of the field roughly speaking in the absence ofgravitational fields tensor119872119901

119894119896119897is not equal to zero) Although

the formula (34) gives a qualitative representation of thegeometric structure it is a little convenient since it reentersthe values of Γ119894

119895119896

Further we establish the equation which is similar toequation of Ricci-Jacobi

119877119901

119894119896119897+ 119877119901

119896119897119894+ 119877119901

119897119894119896= 119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896+ Γ119901

119902119896119878119902

119897119894+ Γ119901

119902119896119878119902

119894119896+ Γ119901

119902119894119878119902

119896119897

= 119878119901

119894119896119897+ Γ119902

119894119897119878119901

119902119896+ Γ119902

119896119897119878119901

119894119902+ 119878119901

119896119897119894+ Γ119902

119896119894119878119901

119902119897+ Γ119902

119897119894119878119901

119896119902

+ 119878119901

119897119894119896+ Γ119902

119897119896119878119901

119902119894+ Γ119902

119894119896119878119901

119897119902

= 119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896

+ 119878119901

119897119902119878119902

119894119896+ 119878119901

119896119902119878119902

119897119894+ 119878119901

119894119902119878119902

119896119897

(36)

It is easy to prove the equations

119878119894

119895119901119878119901

119896119894+ 119878119894

119896119901119878119901

119894119895= 0

119878119894

119894119901119878119901

119895119896= 0

(37)

and as a consequence we obtain the equation

119878119894

119895119901119878119901

119896119894+ 119878119894

119896119901119878119901

119894119895+ 119878119894

119894119901119878119901

119895119896= 0 (38)

3 The Study of Geodesics inthe Space with Torsion

Formula (13) consists of three summand of various kindsThe sum (14) is a geometric object second valence its

components are converted by a formula similar to the one thattakes place for the connection

1198751199011015840

11989610158401198971015840= 119875119901

119896119897

1205971199091199011015840

120597119909119901

120597119909119896

1205971199091198961015840

120597119909119897

1205971199091198971015840+

1205972

119909119901

1205971199091198961015840

1205971199091198971015840

1205971199091199011015840

120597119909119901 (39)

but note that in this space this ldquoconnectionrdquo does not satisfythe ldquotorsion conditionrdquo (119878119894

119895119896equiv Γ119894

119895119896minus Γ119894

119896119895)


Value (12)119878119901119896119897 this tensor can be to be used as a ldquocon-

nectionrdquo but then this ldquoconnectionrdquo is not used to satisfy thecompatibility condition

Value

119872119901

119896119897equiv

1

2

119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) (40)

is a tensor that expresses the combined effect of the metricand torsion

First of all it is easy to show that the element (12)119878119901119896119897does

not affect the geodesic since by the asymmetry of the torsionit is not included in the equations of geodesic lines

1198892

119909119896

1198891205912= minusΓ119896

119894119895

119889119909119894

119889120591

119889119909119895

119889120591

(41)

that is (6) will be determined only by sum 119875119901

119896119897+119872119901

119896119897 In (6) 120591

is the canonical parameter that is the vector 119889119909119894119889120591 is paralleltransported vector For no isotropic geodesic the length of arc119904 is a canonical parameter for geodetic related to 119904 takes placein differential equations

1198892

119909119896

1198891199042

= minusΓ119896

119894119895

119889119909119894

119889119904

119889119909119895

119889119904

(42)

Definition 5 The line is called a geodesic if any tangent to thisline at some point vector remains tangent to it at the paralleltransport along it

Theorem 6 The equations of geodesic lines in space 119884119899 is

determined by the geometric object in form of the sum

119875119901

119896119897+119872119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894)

+

1

2

119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

(43)

In case classical Riemannian space geodetic lines haveknown extreme properties in this case analogical propertiesof geodesic require additional research

Thus for a no isotropic geodesic with canonical parame-ter arc length 119904 we have differential equations

1198892

119909119896

1198891199042

= minusΓ119896

119894119895

119889119909119894

119889119904

119889119909119895

119889119904

(44)

Consider the problem of calculating the variation of thearc length

Let a nonisotropic curve 119909119894

= 119909119894

(119905) 119905 isin [1199051 1199052] We

calculate the variation of length 120575119878 of the curve 119878

120575119878 = int

1199052

1199051

120575radic119892119894119895

119889119909119894

119889119905

119889119909119895

119889119905

119889119905

= int

1199052

1199051

120575 (119892119894119895(119889119909119894

119889119905) (119889119909119895

119889119905))

2radic119892119894119895(119889119909119894119889119905) (119889119909

119895119889119905)

119889119905

120575 (119892119894119895

119889119909119894

119889119905

119889119909119895

119889119905

) = 119892119894119895119863

119889119909119894

119889119905

119889119909119895

119889119905

+ 119892119894119895

119889119909119894

119889119905

119863

119889119909119895

119889119905

= 2119892119894119895

119889119909119894

119889119905

119863

119889119909119895

119889119905

119863

119889119909119895

119889119905

= 120575

119889119909119895

119889119905

+ Γ119895

119901119896

119889119909119901

119889119905

120575119909119896

119863

120575119909119895

119889119905

=

119889

119889119905

120575119909119895

+ Γ119895

119896119901

119889119909119896

119889119905

120575119909119901

119863

119889119909119895

119889119905

= 119863

120575119909119895

119889119905

+ 119878119895

119901119896

119889119909119901

119889119905

120575119909119896

(45)

where119863 denotes the absolute differential through the param-eter curves of the family at a constant value 119905 and119863 is absolutedifferential charge small displacement 119889119905 curve at a constantparameter of the family then

120575(119892119894119895

119889119909119894

119889119905

119889119909119895

119889119905

) = 2119892119894119895

119889119909119894

119889119905

(119863

120575119909119895

119889119905

+ 119878119895

119901119896

119889119909119901

119889119905

120575119909119896

)

120575119904 = int

1199052

1199051

119892119894119895

119889119909119894

119889119904

119863120575119909119895

+ int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119904

119889119909119901

120575119909119896

= int

1199052

1199051

119863(119892119894119895

119889119909119894

119889119904

120575119909119895

)

minus int

1199052

1199051

119892119894119895119863

119889119909119894

119889119904

120575119909119895

+ int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119904

119889119909119901

120575119909119896

(46)

if the ends of the variable curve are fixed then

120575119904 = int

1199052

1199051

(119892119894119895119878119895

119901119896

119889119909119894

119889119905

119889119909119901

120575119909119896

minus 119892119894119895119863

119889119909119894

119889119905

120575119909119895

) (47)

if the considered curve has fixed length (analytically 120575119904 = 0)then we obtain

int

1199052

1199051

(119892119894119895119878119895

119901119896

119889119909119894

119889119904

119889119909119901

120575119909119896

minus 119892119894119895119863

119889119909119894

119889119904

120575119909119895

) = 0 (48)

Using the fundamental lemma calculus of variations itfollows that

119892119894119896119878119896

119901119895

119889119909119894

119889119904

119889119909119901

minus 119892119894119895119863

119889119909119894

119889119904

= 0 (49)

This equation means that the tangent vector 120585119902 of thecurve is transported according to the law119863120585

119902

119892119895119902

119892119894119896119878119896

119901119895120585119894

119889119909119901

which means that 119904 is not geodesic curve Conversely thevariation of the length of the geodesic lines is

120575119904 = int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119905

119889119909119901

120575119909119896

(50)

Properties in the new geodesic geometry defined bymeans of two tensors 119892

119894119896and 119878

119895

119894119896differ greatly from similar

properties in Riemann geometry The theorem is proved


Theorem 7 In order a line in space 119884119899 (that is generated by119892119894119896and 119878

119895

119894119896) was geodetic it is necessary and sufficient that a

variation of the arc this line was equaled

int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119905

119889119909119901

120575119909119896

(51)

Consequence In the case of spaces with affine connectionis known that there is a gap (breach) of closure during thetransition from the original to the image and vice versa inthe case of an infinitely small contour (determined up to thesecond order relative to 120591) If you specify the torsion tensor 119878119896

119894119895

at the corresponding point then if this gap is denoted by Ψ119896then Ψ

119896

= 119878119896

119894119895119860119894

119861119895

1205912 where the parallelogram 119860

119894

120591 and 119861119895

120591

shrink to a point at 120591 rarr 0 In this case we have the formulafor square of the length |Ψ|2 = 119892

119901119902119878119901

119894119895119860119894

119861119895

119878119902

119896119897119860119896

119861119897

1205914

Theorem 8 Let classical Riemannian space be given (Rie-mannian manifold with Riemannian metric tensor 119892

119894119896) with

the connection 119875119896

119894119895 connection Riemannian space Let 119884119899 be

the space generated jointly and agreed on by metric 119892119894119896and

torsion 119878119896119894119895tensors together with connection Γ119896

119894119895 To coincide the

geodesics in classical Riemannian space with the geodesics inspace 119884119899 it is necessary and sufficient that the connections 119875119896

119894119895

and Γ119896119894119895be differed by tensor

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (52)

that is

Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (53)

Remark 9 The last summand in

Γ119896

119894119895=

1

2

119892119896119901

(119892119894119901119895119897

+ 119892119895119901119894

minus 119892119894119895119901

+ 119892119894119898119878119898

119895119901+ 119892119895119898119878119898

119894119901)

+

1

2

119878119896

119894119895

(54)

that is (12)119878119896119894119895 is irrelevant (it is clear)

Proof The necessity means that if the geodesics in classicalRiemannian space coincide with the geodesics in space 119884119899then

Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (55)

Let the vector 119860119894 be tangent to any geodesic and 119860119894 is

parallel transported to the connection Γ119901119896119897

Γ119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

+ 119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

+

1

2

119878119901

119896119897

(56)

and a vector 119861119894 is parallel transported to the connection 119875119901119896119897

119875119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894) (57)

It was found that the first geodesic connections coincidewith geodesic connection 119875

119901

119896119897+ 119872119901

119896119897 here 119872119896

119894119895is an arbitrary

symmetric tensor Since both vectors are tangential then

119861119894

= 119886119860119894

(58)

where the coefficient 119886 is variable and 119886 = 0 Tensor-vector119860119894 is given by

119889119860119896

= minus (119875119896

119894119895+119872119896

119894119895)119860119894

119889119909119895

119889119861119896

= minus119875119896

119894119895119861119894

119889119909119895

(59)

Then we have

119860119896

119889119886 + 119886119889119860119896

= minus119875119896

119894119895119886119860119894

119889119909119895

(60)

119860119896

119889119886

119886

= 119872119896

119894119895119860119894

119889119909119895

(61)

Tangent vector119860119894 can be written119860119894 = 119889119909119894

119889120591 where 120591 is thecanonical parameter relative to the connection 119875119901

119896119897+119872119901

119896119897

Then after dividing by 119889120591 we have

119889 ln 119886119889120591

119860119896

= 119872119896

119894119895119860119894

119860119895

(62)

Since geodesic lines can be carried out through any point andin any direction then equality must be true at any point andfor any vector119860119894 the functional dependence of the point anddirection obviously there

The last equality is multiplied by 119860119897 and alternated by 119896and 119897

119860119897

119872119896

119894119895119860119894

119860119895

minus 119860119896

119872119897

119894119895119860119894

119860119895

= 0 (63)

or

120575119897

119898119872119896

119894119895119860119894

119860119895

119860119898

minus 120575119896

119898119872119897

119894119895119860119894

119860119895

119860119898

= 0 (64)

where we denote 120575119897

119898= 119892119898119901119892119901119897 This equation must hold

identically with respect to vectors 1198601 119860119899 consequentlyafter summed similar summands all the coefficients of thecubic form must vanish We compute the total coefficient

120575119897

119898119872119896

119894119895minus 120575119896

119898119872119897

119894119895+ 120575119897

119894119872119896

119895119898minus 120575119896

119894119872119897

119895119898+ 120575119897

119895119872119896

119898119894minus 120575119896

119895119872119897

119898119894

= 0

(65)

Then we contracted tensor by indices 119897 and 119895 Since 120575119894119894= 119899 we

have

119872119896

119894119895=

1

119899 + 1

(120575119896

119894119872119897

119895119897+ 120575119896

119895119872119897

119894119897) (66)


All calculations above are correct for any symmetricaltensor119872119896

119894119895

All calculations presented above do not take intoaccount the specificity of the tensor 119872119896

119894119895 then let 119872119901

119896119897equiv

(12)119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) then119872119897

119896119897equiv (12)119878

119897

119896119897

We obtained

Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (67)

The necessity is proof

Remark 10 In the beginning we considered more generalsituation and then specified tensor119872119896

119894119895

We prove the sufficiency We assume that Γ119896119894119895minus 119875119896

119894119895=

(12)(1(119899 + 1))(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) we have to show that the

geodesics coincideWe again use (61) we have 119860119896119889119886119886 = 119872

119896

119894119895119860119894

119889119909119895


119889120591 where 120591 isthe canonical parameter relative to the connection Γ119896

119894119895

Since along the curve 119878119897119894119897119860119894 there is a definite function of

the parameter 120591 then it will find ln 119886 after integration withup to a constant 119886 but only up to a constant factor (we canintegrate) Therefore the vector is found to be 119861119894 = 119886119860

119894 andall geodesics coincide The sufficiency is proved The theoremis proved

4 The Theory of Hypersurfaces

First we formulate a well-known classical result on thederivation formulas of hypersurface in Riemannian torsion-free space where the connection is uniquely generated by themetric

Formal statement of Gauss-Codazzi equations in spacewith symmetrical connection 119878119896

119894119895= 0 Assume that 119894 119872 sub 119875

is 119899-dimensional embedded submanifold of a Riemannianmanifold 119875 of dimension 119899 + 119901 There is a natural inclusionof the tangent bundle of119872 into that of 119875 by the pushforwardand the cokernel is the normal bundle of 119872 0 rarr 119879

119909rarr

119879119909119875|119872

rarr 119879perp

119909119872 rarr 0 The metric splits this short exact

sequence and so 119879119875|119872= 119879119872 oplus 119879

perp

119872The Levi-Civita connection nabla

1015840 of 119875 decomposes intotangential and normal components For each 119883 isin 119879119872 andvector field 119884 on 119872 nabla1015840

119883119884 = 119879(nabla

1015840

119883119884) + perp (nabla

1015840

119883119884) Let it be

nabla119883119884 = 119879(nabla

1015840

119883119884) 120572(119883 119884) = perp (nabla

119883119884) Gaussrsquo formula asserts

that nabla119883is Levi-Civita connection on119872 and 120572 is a symmetric

form with values into the normal bundle It is also referred toas the second fundamental form An immediate corollary isthe Gauss equation For119883119884 119885119882 isin 119879119872

⟨1198771015840

(119883 119884)119885119882⟩ = ⟨119877 (119883 119884)119885119882⟩

+ ⟨120572 (119883 119885) 120572 (119884119882)⟩

minus ⟨120572 (119884 119885) 120572 (119883119882)⟩

(68)

where 1198771015840 is the Riemann curvature tensor of 119875 and 119877 is thatof119872

There are thus a pair of connections nabla defined on thetangent bundle of 119872 and 119863 defined on the normal bundleof 119872 These combine to form a connection on any tensorproduct of copies of 119879119872 and 119879 perp 119872 In particular theydefined the covariant derivative of 120572

(nabla10158401015840

119883120572) (119884 119885) = 119863

119883(120572 (119884 119885)) minus 120572 (nabla

119883119884119885)

minus 120572 (119884 nabla119883119885)

(69)

Codazzi equation is

perp ⟨1198771015840

(119883 119884)119885⟩ = (nabla10158401015840

119883120572) (119884 119885) minus (nabla

10158401015840

119884120572) (119883 119885) (70)

The above formulas also hold for immersions becauseevery immersion is in particular a local embedding Theimportant assumption in the theory stated above is thatconnection is symmetrical so analog of second form ofhypersurface is symmetrical and we could obtain the deriva-tion formulas of hypersurface

In our case we do not assume any conditions aboutthe connection symmetry so below we study much morecomplicated problem

A similar manner as in the Riemannian space can bedeveloped as a theory of hypersurfaces so in themetric spacewith torsion a theory of hypersurfaces can be constructedBut due to the presence of torsion in these cases there is asignificant difference For example the derivation equations(analog PetersonCodazzi equations) take amore complicatedform in which there are new summands which are caused bythe presence of torsion in the space

We make a study of the hypersurfaces 119884119899minus1 in a metricspace with torsion119884119899 We are assuming that the hypersurfaceis defined by a system of equations

119909119894

= 119909119894

(1199101

119910119899minus1

) (71)

and the rank of the matrix [120597119909119894120597119910120572] equals 119899 minus 1 The metrictensor of hypersurface 119884119899minus1 is given by

119886120572120573

= 119892119894119895

120597119909119894

120597119910120572

120597119909119895

120597119910120573

(72)

Then we obtain the formula for tensor of torsion 119879120574

120572120573of

hypersurface 119884119899minus1 (assuming that functions 119909119894(1199101 119910119899minus1)are smooth enough) Let 119866120572

120573120574be the connection of 119884119899minus1 and

we assume that 119866120572120573120574

express via metric 119886120572120573


120572120573

similarly to as the connection Γ119896119894119895express by means of 119892

119894119895and

119878119896

119894119895 we have

119866120572

120573120574

=

1

2

(119886120572120578

(119886120573120578120574

+ 119886120574120578120573

+ 119886120573120574120578

+ 119886120573120583119879120583

120574120578+ 119886120574120583119879120583

120573120578) + 119879120572

120574120573)

(73)


119866120572

120573120574is geometric object and is subjected to the law of the

transformation from one coordinate system 119906120572 to another 119906120572

1015840

by the formula

119866120572

120573120574= 1198661205721015840

12057310158401205741015840

120597119906120572

1205971199061205721015840

1205971199061205731015840

120597119906120573

1205971199061205741015840

120597119906120574+

120597119906120572

1205971199061205721015840

1205972

1199061205721015840

120597119906120573120597119906120574

(74)

Then we assume that connection 119866120572

120573120574of 119884119899minus1 are associated

with connection Γ119896119894119895of 119884119899 mean of formula

119866120572

120573120574

120597119909119896

120597119906120572= Γ119896

119894119895

120597119909119894

120597119906120573

120597119909119895

120597119906120574+

1205972

119909119896

120597119906120573120597119906120574

(75)

We obtain

1

2

(119886120572120578

(119886120573120578120574

+ 119886120574120578120573

+ 119886120573120574120578

+ 119886120573120583119879120583

120574120578+ 119886120574120583119879120583

120573120578) + 119879120572

120574120573)

120597119909119896

120597119906120572

=

1

2

(119892119896119899

(119892119899119894119895

+ 119892119899119895119894

minus 119892119894119895119899

+ 119892119894119898119878119898

119895119899+ 119892119895119898119878119898

119894119899) + 119878119896

119894119895)

times

120597119909119894

120597119906120573

120597119909119895

120597119906120574+

1205972

119909119896

120597119906120573120597119906120574

(76)

By permuting indices we have the next formula for thetorsion tensor of hypersurface 119884119899minus1

119879120574

120572120573= 119886120574120578

119892119901119902119878119901

119894119895

120597119909119894

120597119910120572

120597119909119895

120597119910120573

120597119909119902

120597119910120578 (77)

using tensors 119886120572120573

and 119879120574

120572120573 both metric and torsion we can

explore the geometry of the space hypersurface 119884119899minus1The connection of119884119899minus1 will be determined by the formula

(73)Below we use the mixed tensors values which enumer-

ated two types of indices while Latin indices refer to thecontaining space 119884

119899 and are responsive to the coordinatetransformation 119909

119894 and Greek indices belong to the spacehypersurface 119884

119899minus1 and are responsive to the coordinatetransformation 119910120572

The index 119894 is not responsive to the coordinate 119910120572

transformation into 119884119899minus1 and the index 120572 does not respondto the coordinate 119909

119894 transformation in 119884119899 For example

the formula to calculate the covariant derivative of a mixedtensor

119860119894120572

119895120573120574= 119860119894120572

119895120573120574+ Γ119894

119901119896119860119901120572

119895120573

120597119909119896

120597119910120574minus Γ119901

119895119902119860119894120572

119901120573

120597119909119902

120597119910120574+ 119866120572

120578120574119860119894120578

119895120573

minus 119866120578

120573120574119860119894120572

119895120578

(78)

The direct calculations lead us to formulas

119906119894120572120573

minus 119906119894120573120572

= 119877119901

119896119897119894119906119901

120597119909119897

120597119910120572

120597119909119896

120597119910120573

+ 119878119902

119896119897119906119894119902

120597119909119897

120597119910120572

120597119909119896

120597119910120573

119906120574120572120573

minus 119906120574120573120572

= 119877120578

120573120572120574119906120578+ 119879120578

120573120572119906120574120578

(79)

where 119877120578

120573120572120574is curvature tensor of space 119884

119899minus1 compiled byusing the components of connection 119866120572

120573120574

A further aim of our study is to obtain some analogs ofPeterson-Kodachi equations To do this consider the systemof values

120585119894

120572=

120597119909119894

120597119910120572 (80)

At each point of the hypersurface 119884119899minus1 we can build

rapper consisting of the vectors

120585119894

1 120585

119894

119899minus1 ]119894 (81)

where 120585119894

1 120585

119894

119899minus1are linearly independent tangent vectors

and ]119894 is normal vector defined since the metric and connec-tivity agreed

Next we act formally the idea is the same as in theclassical case and we will indicate significant new momentsWe compute the derivative of the mixed tensors 120585119894

120572

120585119894

120572120574= 120585119894

120572120574+ Γ119894

119901119902120585119901

120572

120597119909119902

120597119910120574minus 119866120578

120572120574120585119894

120578 (82)

In contrast to the case of torsion-free connection we havethe equality

120585119894

120572120574minus 120585119894

120574120572= 119878119894

119901119902120585119901

120572120585119902

120574+ 119879120578

120574120572120585119894

120578 (83)

Next we permute the indices in equation

0 = 119886120572120573120574

= (119892119894119895120585119894

120572120585119895

120573)

120574

= 119892119894119895120585119894

120572120574120585119895

120573+ 119892119894119895120585119894

120572120585119895

120573120574 (84)

we obtain

119892119894119895120585119894

120572120574120585119895

120573= 0 (85)

Hence we can write a decomposition

120585119894

120573120572= 120587120572120573]119894 (86)

Remark 11 120587120572120573

is tensor which is similar to the secondfundamental tensor of hypersurfaces 119884119899minus1 but its structurein this space is substantially different from the case ofRiemannian spaces with zero torsion

Then we have obtained (by differentiating 119892119894119895]119894120585119895120572= 0 by

120574) the formula

119892119894119895]119894120574120585119894

120572= minus120587120574120572 (87)

Similarly by differentiating 119892119894119895]119894]119895 = 1 by 120574 we obtain

]119894120574= minus119886120578120583

120587120583120574120585119894

120578 (88)

Formulae (86) and (88) characterize the change of vectorsin the small accompanying frame relative to this frame itself


Further we obtain

120585119894

120573120594120582minus 120585119894

120573120582120594= minus119877119894

119896119897119901120585119896

120582120585119897

120594120585119901

120573+ 119877120590

120582120594120573120585119894

120590+ 119879120590

120582120594120585119894

120573120590

= (120587120594120573120582

minus 120587120582120573120594

) ]119894

minus (120587120594120573120587120578120582119886120578120590

minus 120587120582120573120587120578120594119886120578120590

) 120585119894

120590

(89)

Equation (89) is multiplied by 119892119894119895120585119895

120572 we have

119877120572120582120594120573

= 119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573120585119894

120572minus (120587120594120573120587120572120582

minus 120587120582120573120587120572120594) (90)

Similarly we derive a formula

]119894120594120582

minus ]119894120582120594

= minus119877119894

119896119897119901120585119896

120582120585119897

120594]119901 + 119879

120590

120582120594]119894120590

= (120587120578120582120594

119886120578120590

minus 120587120578120594120582

119886120578120590

) 120585119894

120590

(91)

We contract (89) with 119892119894119895]119895 then

minus119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573]119894 + 119879

120590

120582120594120587120590120573

= 120587120594120573120582

minus 120587120582120573120594

(92)

Formula (91) is multiplied by 119892119894119895120585119895

120572 we concluded that

minus119877119894119896119897119901

120585119896

120582120585119897

120594]119901120585119894120572+ 119879120590

120582120594120587120572120590

= 120587120572120582120594

minus 120587120572120594120582

(93)

Remark 12 If (91) contract with 119892119894119895]119895 then we obtain identi-

cally zero equals zeroThus we have the two types of formulas Formula (90)

does not contain the torsion tensor explicitly but it is countedin the tensor 120587

120572120573 In the formula (91) the torsion tensor

of the hypersurface is present explicitly and in the formof coefficients of 120587

120572120573and appears in the calculation of the

covariant derivative

5 Establish Some Important Relationships

We consider the equation

119878119894

119895119896119901119902minus 119878119894

119895119896119902119901= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

+ 119878119905

119902119901119878119894

119895119896119905

(94)

or

119878119894

119895119896119901119902minus 119878119894

119895119896119902119901minus 119878119905

119902119901119878119894

119895119896119905

= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

(95)

we contract these tensors by indices 119894 119902 then the left side ofthis equation can be transformed into

119878119894

119895119896119901119894minus 119878119894

119895119896119894119901minus 119878119905

119894119901119878119894

119895119896119905

= (119878119894

119895119896119901minus 119878119894

119902119901119878119902

119895119896)

119894

minus 119878119894

119895119896119894119901minus 119878119894

119901119902119894119878119902

119895119896

(96)

then we contract this equation by indices 119896 119901 and raising theindex 119895 we obtained

(119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896)119894

minus 119892119896119901

119892119895119904

119878119894

119904119896119894119901minus 119892119896119901

119892119895119904

119878119894

119901119902119894119878119902

119904119896

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905

minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(97)

we introduce the notation

119867119895119894

= 119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896

119865119895119901

= 119892119896119901

119892119895119904

119878119894

119904119896119894

(98)

Then without any loss of generality we obtain therelations

119867119895119894

119894minus 119865119895119894

119894minus 119892119896119901

119892119895119904

119878119902

119904119896119865119901119902

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(99)

where 119865119901119902

= 119878119894

119901119902119894

Suppose now that the identity Ricci-Jacobi run in astandard form 119877119901

119894119896119897+ 119877119901

119896119897119894+ 119877119901

119897119894119896= 0 hence

119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896+ 119878119901

119897119902119878119902

119894119896+ 119878119901

119896119902119878119902

119897119894+ 119878119901

119894119902119878119902

119896119897= 0 (100)

We contract this equation by indices 119901 119897 and we foundidentity

119878119901

119894119896119901+ 119878119901

119896119901119894+ 119878119901

119901119894119896= 0 (101)

Next we are assuming that 119878119901119894119901

= 120593119894and taking into

account the identity 119878119901

119894119895119878119902

119901119902= 0 we obtain the following

expression

119878119901

119894119895119901= 120593119894119895minus 120593119895119894 (102)

Next if we put 119878119901119894119895119901

= 0 then it follows that 120593119894119895minus 120593119895119894

=

0 and hence the value 119878119901119894119901can be expressed in terms of the

partial derivative of the scalar 119878119901119894119901= 120593119894= (ln120595)

119894 System (99)

takes the form

119867119895119894

119894= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

119865119894119895

= 0

(103)

We consider the tensor

119862119894119895119896

= 119892119901119895

119892119902119896

119878119894

119901119902+ 119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896

119901119902 (104)

obvious that tensor is antisymmetricWe have the equality

119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+ 119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904

minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

(105)


By direct calculations we can conclude that

119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

=

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

(106)

hence

119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902) (107)

We calculate the covariant derivative

119862119894119896119895

119894= minus119862119894119895119896

119894= minus (119862

119894119895119896

119894+ Γ119895

119901119894119862119894119901119896

+ Γ119896

119901119894119862119894119895119901

+ Γ119894

119901119894119862119901119896119895

) (108)

By virtue of the fact that tensor 119862119894119895119896

= 119892119901119895

119892119902119896

119878119894

119901119902+

119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896

119901119902is antisymmetric we have

Γ119895

119901119894119862119894119901119896

= Γ119895

119894119901119862119894119901119896

=

1

2

(Γ119895

119894119901119862119894119901119896

+ Γ119895

119901119894119862119901119894119896

) =

1

2

119878119895

119894119901119862119901119896119894

= minus

1

2

119878119895

119901119894119862119896119901119894

(109)

similarly we obtain

Γ119896

119901119894119862119894119895119901

= Γ119896

119894119901119862119901119895119894

=

1

2

119862119895119901119894

(Γ119896

119894119901minus Γ119896

119901119894) =

1

2

119878119896

119894119901119862119895119901119894

=

1

2

119878119896

119901119902119862119895119902119901

(110)

Then we write

119862119894119896119895

119894= minus119862119894119895119896

119894

= minus119862119894119895119896

119894minus

1

2

119878119895

119901119902119862119896119901119902

+

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894+

1

2

119878119895

119901119902119862119896119901119902

minus

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

(111)

We will compute Γ119901

119897119901 for this we recall that Γ

119901

119901119897=

(12)119892119894119901119897119892119894119901

= (1radic119892)(120597radic119892120597119909119897

) and Γ119901

119897119901= Γ119901

119901119897+ 119878119901

119897119901and

obtain

Γ119901

119901119897=

1

radicminus119892

120597radicminus119892

120597119909119897

+ (ln120595)119897= (ln (120595radicminus119892))

119897 (112)

Then we obtain

119867119895119896

minus 119867119896119895

minus 119865119895119896

= minus119862119894119895119896

119894minus (ln (120595radicminus119892))

119894119862119894119896119895

(113)

We multiply by 120595radicminus119892 and have

120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)

= minus120595radicminus119892 (119862119894119895119896

119894+ (ln (120595radicminus119892))

119894119862119894119896119895

)

120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)

= minus (120595radicminus119892119862119894119895119896

)119894

(114)

We differentiate the last equality in view of the anti-symmetry of the tensors and we obtain the next importantequality

(120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

))119896

= 0 (115)

6 The Field Equations

Below we consider the derivation of the field equations independing on conditions

61 The Field Equations in the Absence of Symmetry Con-ditions Let Γ119894

119895119896be constructed on the basis of connection

119899-dimensional manifold It is not assumed that the Γ119894119895119896

aresymmetric in 119895 and 119896 Regardless of connection Γ

119894

119895119896 we

introduce symmetric metric tensor 119892119894119896(4)

We will derive the field equations from the variationprinciple of least action by varying the function Γ

119894

119895119896and 119892

119894119896

independentlyUsing (23) we can write the Riemann tensor

119877119894119895= Γ119901

119894119901119895minus Γ119901

119894119895119901+ Γ119901

119902119895Γ119902

119894119901minus Γ119901

119902119901Γ119902

119894119895 (116)

We form the scalar density as (119877119894119896+119878119895

119894119897119878119897

119896119895)119892119894119896

radicminus119892 where 119878119894119895119896=

Γ119894

119895119896minus Γ119894

119896119895 and postulate that all the variations of the integral

int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881 (117)

with respect to Γ119894119895119896and 119892119894119896radicminus119892 as the independent variables

are zero (at the boundaries they do not vary)Without dwelling on the standard intermediate calcula-

tions we find that the variation with respect to 119892119894119896radicminus119892 leadsto the equation

119877119894119896+ 119878119895

119894119897119878119897

119896119895= 0 (118)

then the variation with respect to Γ119894119895119896gives the equation

119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901

minus 120575119895

119896(119892119894119901

119901minus

1

2

119892119894119901

119892119898119899119892119898119899

119901+ 119892119901119902

Γ119894

119901119902) minus 119892119894119895

Γ119901

119896119901

+ 2 (119892119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(119)


If we contract the left side of the last equality by indices119894 and 119896 then we obtain zero identically equal to zero If wecontract by indices 119895 and 119896 we obtain the equation

minus3119892119894119895

119895+

3

2

119892119894119895

119892119901119902119892119901119902

119895minus 3119892119901119902

Γ119894

119901119902+ 119892119894119901

119878119895

119901119895= 0 (120)

Therefore we have

119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901minus 119892119894119895

Γ119901

119896119901

minus

1

3

120575119895

119896119892119894119901

119878119902

119901119902+ 2 (119892

119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(121)

Then we lowered upper indices by using the metric andobtain the equation

minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901+ 2 (119892

119901119895119878119901

119894119896+ 119892119894119901119878119901

119896119895) = 0

(122)

We use the symmetry of tensor 119892119894119895and rearrange 119894 and 119895

and deduce the equation

5 (119892119894119901119878119901

119896119895+ 119892119895119901119878119901

119894119896) +

1

3

(119892119895119896119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0 (123)

Further we obtained

minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901minus

2

15

(119892119896119895119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0

(124)

Thus we obtained the field equations (122) by varyingconnection obtained equation and as a result of them weobtained (123) where there are only metric and torsion Weassuming that the electromagnetic component is absent 120593 equiv

0 then from (123) we have 119878119894

119895119896equiv 0 and connection is

symmetrical as in Riemann geometry and (124) shows theknown law of Einstein-Hilbert problem for the gravitationalfield If 120593 = 0 and the metric is flat (no gravitationalfield) then from (124) can be obtainedMaxwell equations forelectromagnetic field in vacuum

62 The Einstein-Hilbert Equation in Case of the Absenceof Symmetry Conditions When the Lagrange Function IsDepending on the Torsion Tensor Now we also start from thevariation principle of the least action in the form 120575(119882

119898+

119882119892) = 0 where 119882

119898and 119882

119892-action respectively for matter

and field values we are varying 119892119894119896

By standard calculations we have

120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int (119877119894119896radicminus119892120575119892

119894119896

+ 119877119894119896119892119894119896

120575radicminus119892 + 119892119894119896

radicminus119892120575119877119894119896

+ 119878119895

119894119897119878119897

119896119895radicminus119892120575119892

119894119896

+ 119878119895

119894119897119878119897

119896119895119892119894119896

120575radicminus119892) 119889119881

119877119894119896119892119894119896

120575radicminus119892 = minus

1

2

119877119901119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(125)

Similarly we obtain

119878119895

119894119897119878119897

119896119895119892119894119896


1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(126)

Nowwe compute119892119894119896radicminus119892120575119877119894119896directly by using the defini-

tion and thus obtain two types of summands the first has thestandard form 119892

119894119896

(120575Γ119897

119896119894)119897minus 119892119894119896

(120575Γ119897

119896119897)119894= (119892119894119896

120575Γ119897

119896119894minus 119892119894119897

120575Γ119901

119897119901)119897

where it is considered that 119892119894119896119897

= 0 and by Stokesrsquo theoremturns into zeros The term of the second type exists due tothe absence of symmetry connection 119892119894119896120575119877

119894119896= 119892119894119896

120575(Γ119901

119902119901Γ119902

119896119894minus

Γ119901

119902119894Γ119902119897

119896119901) Then we express the connection coefficients via the

metric and torsion and after a rather lengthy calculation weobtain

119892119894119896

120575119877119894119896= (119878119901

119894119901119878119902

119902119896minus 119878119901

119894119902119878119902

119901119896minus 119892119896119901119892119902119905

119878119898

119894119905119878119901

119902119898) 120575119892119894119896

(127)

Thus we have

120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(128)

Then we obtain the conclusions

120575119882119892

= 1198701int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= 1198701int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(129)

where a physical constant 1198701 as a rule in a classic case is

1198883

16120587119896 and 119896 is called the universal gravitational constantFor variation of the action of matter we find

120575119882119892= 1198702int119879119894119896radicminus119892120575119892

119894119896

119889119881 (130)

where 119879119894119896is energy-momentum tensor of matter 119870

2usually

take a constant equal to minus12119888Therefore by the principle of least action for 120575119882

119892+120575119882119898=

0 we find relations

int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896minus 119870119879119894119896)radicminus119892120575119892

119894119896

119889119881 = 0

(131)


because of the arbitrariness 120575119892119894119896 we have

119877119894119896minus

1

2

119892119894119896119877 + 119878119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902+ 119878119895

119894119897119878119897

119896119895

minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896= 119870119879119894119896

(132)

where the constant119870 can be determined by 1198701and119870

2

7 Conclusions

We have investigated the properties of the space generatedjointly and agreed on by the metric and the torsion tensorsWe have presented the structure of the curvature tensor andstudied its special features and for this tensor we obtainedanalog Ricci-Jacobi identity and also evaluated the gap thatoccurs at the transition from the original to the imageand vice versa in the case of an infinitely small contoursThe geodesic lines equation has been researched We haveshown that the structure of tensor 120587

120572120573 which is similar

to the second fundamental tensor of hypersurfaces 119884119899minus1 issubstantially different from the case of Riemannian spaceswith zero torsion Then we have obtained formulas forhypersurfaces 119884119899minus1 which characterize the change of vectorsin accompanying basis relative to this basis itself

Taking into consideration the structure of the space withmetric and torsion we have reach the aim of our paperthat is derived from the variation principle the generalfields equations (electromagnetic and gravitational) so we areobtained analog of Einstein-Hilbert equation in space 119884119899

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] K Bredies ldquoSymmetric tensor fields of bounded deformationrdquoAnnali di Matematica Pura ed Applicata vol 192 no 5 pp 815ndash851 2013

[2] E Cartan and J Schouten ldquoOn Riemannian geometries admit-ting an absolute parallelismrdquoNederlandse Akademie vanWeten-schappen Proceedings Series A vol 29 pp 933ndash946 1926

[3] I Agricola and T Friedrich ldquoOn the holonomy of connectionswith skew-symmetric torsionrdquo Mathematische Annalen vol328 no 4 pp 711ndash748 2004

[4] I Agricola and T Friedrich ldquoA note on flat metric connectionswith antisymmetric torsionrdquo Differential Geometry and ItsApplications vol 28 no 4 pp 480ndash487 2010

[5] B Alexandrov and S Ivanov ldquoVanishing theorems on Hermi-tian manifoldsrdquo Differential Geometry and its Applications vol14 no 3 pp 251ndash265 2001

[6] A Saa ldquoA geometrical action for dilaton gravityrdquo Classical andQuantum Gravity vol 12 no 8 pp L85ndashL88 1995

[7] G Bonneau ldquoCompact Einstein-Weyl four-dimensional mani-foldsrdquo Classical and Quantum Gravity vol 16 no 3 pp 1057ndash1068 1999

[8] E Cartan and J Schouten ldquoOn the geometry of the groupman-ifold of simple and semisim-ple groupsrdquoNederlandse Akademievan Wetenschappen Series A vol 29 pp 803ndash815 1926

[9] G R Cavalcanti ldquoReduction of metric structures on CourantalgebroidsrdquoThe Journal of Symplectic Geometry vol 4 no 3 pp317ndash343 2006

[10] T Dereli and R W Tucker ldquoAn Einstein-Hilbert action foraxi-dilaton gravity in four dimensionsrdquo Classical and QuantumGravity vol 12 no 4 pp L31ndashL36 1995

[11] A Einstein The Meaning of Relativity Princeton UniversityPress Princeton NJ USA 1921

[12] A Einstein Relativity The Special and General the Ory H Holtand Company New York NY USA 1920

[13] A Einstein ldquoTheorie unitaire de champ physiquerdquo Annales delrsquoInstitut Henri Poincare no 1 pp 1ndash24 1930

[14] S Manoff ldquoFrames of reference in spaces with affine connec-tions and metricsrdquo Classical and Quantum Gravity vol 18 no6 pp 1111ndash1125 2001

[15] R AMosna andA Saa ldquoVolume elements and torsionrdquo Journalof Mathematical Physics vol 46 no 11 Article ID 112502 2005

[16] J A Peacock Cosmological Physics Cambridge UniversityPress Cambridge UK 1999

[17] P J E Peebles Principles of Physical Cosmology PrincetonUniversity Press Princeton NJ USA 1993

[18] W Rindler Essential Relativity Special General and Cosmologi-cal Texts and Monographs in Physics Springer New York NYUSA 2nd edition 1977

[19] J Jost Riemannian Geometry and Geometric Analysis SpringerBerlin Germany 2005

[20] H Pedersen and A Swann ldquoRiemannian submersions four-manifolds and Einstein-Weyl geometryrdquo Proceedings of the Lon-don Mathematical Society Third Series vol 66 no 2 pp 381ndash399 1993

[21] S Dineen Multivariate Calculus and Geometry SpringerUndergraduate Mathematics Series Springer Berlin Germany3rd edition 2014

[22] J G VargasDifferential Geometry for Physicists andMathemati-cians Moving Frames and Differential Forms From Euclid PastRiemann World Scientific 2014

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In applications the statement and definition of action aretaken together

120575int

1199052

1199051

119871 (119902 (119905) 119902 (119905)) 119889119905 = 0 (2)

The action and Lagrangian both contain the dynamics ofthe system for all times The term ldquopathrdquo simply refers to acurve traced out by the system in terms of the coordinates inthe configuration space that is the curve 119902(119905) parameterizedby time On the other hand a Finsler manifold is a differ-entiable manifold together with the structure of an intrinsicquasimetric space in which the length of any rectifiable curve120591 [119886 119887] rarr 119872 is given by the length functional

119878 (120591 (119905)) = int

1199052

1199051

119865 (120591 (119905) 120591 (119905)) 119889119905 (3)

where 119865(119909 sdot) is a Minkowski norm on each tangent space Itis obvious from these definitions that there is a connectionbetween these two concepts which can be realized byHamiltonian formalism Thus any Riemannian space can beregarded as Finsler manifold with the length functional 1198652 =119892119894119895(119909)119889119909

119894

119889119909119895 and so the geodesics of a Finsler manifold are

geodesics of Riemannian spaceThe geodesics of the space that are being studied in

our work (with the geometric structure generated by metric119892119894119895(119909) and torsion 119878119896

119894119895(119909) tensors) are different from geodesics

of corresponding Riemann space (with 119892119894119895(119909)) and so of

geodesics Finsler manifold (with 1198652 = 119892119894119895(119909)119889119909

119894

119889119909119895)

These spaces retain all the properties of geometry ofan affine space but important features associated with thepresence of the metric appear the structure of the curvaturetensor has a specific characteristics as well as an opportunityto assess the gap that occurs at the transportation from theoriginal to the image and conversely in the case of infinitelysmall contours

The main objective of this work-the study of the geomet-ric properties of the space that is generated by metric 119892

119894119895(119909)


119894119895(119909) tensors obtain the field equations from

variation principle in such spacesWe remind that according to Albert Einstein proposal

the free falling gravitating massive bodies follow geodesic lineIf we postulate this proposal we can obtain some results ofNewton theory as a consequenceWe have another importantassumption of Albert Einstein that the geodesic equation ofmotion can be derived from the field equations for emptyspace

Since we believe that gravitational and electromagneticfields are determined geometric structure of empty space(torsion and curvature) it is interesting to have exampleabout geometric sense of torsion

Now we discuss one example about geometric sense oftorsionWe consider the surface 119878 At point119860 on 119878 construct atangent plane 119875 We choose an arbitrary infinitesimal square119860119861119862119863 in the plane 119875 with vertex 119860 From point 119860 on thesurface 119878 will draw the geodesic in the direction of 119860119861 Wepass along the distance corresponding parameter equal to the

length of119860119861 and get to point 1198611015840 Similarly from119860 on 119878 drawgeodesic towards 119860119863 and get into 1198631015840 We perform a paralleltransportation of vector 119860119863 to point 1198611015840 along the geodesic1198601198611015840 and draw geodesic 1198611015840 along the transportation of this

vector we reach the point 1198621015840 Similarly the vector 119860119861 willmove parallel along the1198601198631015840 and along the transported vectorfrom 119863

1015840 draw a geodesic to get to 11986210158401015840 If torsion is zero then1198621015840

= 11986210158401015840 and geodesic square up to small higher order will

be closed otherwise not In our case due to the presence of themetric can calculate the length of the gap more precisely we canestimate the length of this gap

These considerations are true only up to the second orderrelative to the length of square side If we want more strictresult we must consider the component of curvature tensorNext this example is true only when length of square sidetends to zero that is remains very small in other words ingeneral it is a local property

We go to the discussion of physical interpretation of thisexampleThe physical properties of the space-time (more justspace) are defined by the presence of matter (electromagneticfields and mass) in this space and from the viewpoint ofmathematics are described by the geometrical structure ofspace (torsion andmetric tensors)The empty space (withoutmatter) corresponds to the geometric structure of Euclideanspace (torsion tensor and curvature tensor are identicallyequal to zero) Similarly gravitation (mass and without elec-tromagnetic fields) corresponds to the geometric structure ofRiemannian space (torsion tensor is identically equal to zero)And similarly the electromagnetism (electromagnetic fieldsand without mass) corresponds to the geometric structure ofaffine connection space (curvature tensor is identically equalto zero) In the last two cases the result is conditional (notstrict) because the matter division by the mass and field isconditional

To describe the internal geometry of the subspace issufficient to define two tensors the metric tensor 119892

119894119896and the

torsion tensor 119878119898119894119896 using these tensors one can build connec-

tion of the space and all the geometry of the spaceIf we consider the space of hypersurface 119884119899minus1 as a sub-

space then for a complete description of the geometry itis necessary and sufficient to define the tensors 119886

120572120573and

119879120574

120572120573 If we consider the hypersurface 119884

119899minus1 as a subspacethat embedded in a space 119884

119899 then it is not sufficient toknow tensors 119886

120572120573and 119879

120574

120572120573for identifying the space 119884119899 (for

obtaining the metric tensor 119892119894119896and the torsion tensor 119878119898

119894119896)

to describe the geometry of the ambient space it is necessaryand sufficient to specify exactly how the hypersurface 119884119899minus1 isembedded in it that is specify the embedment it can be donein various ways We assume that Jacobi matrix is known sothe way of embedment hypersurface119884119899minus1 in119884119899 is determined(also the space can be recovered by determination of normalfor the hypersurface 119884119899minus1)

We consider the followingmethod of constructing space-time the space is constructed on the basis of manifoldsby determining these manifolds metric tensor and torsiontensorMetric and torsion tensors are calculated from the dif-ferential equations of the field Hence torsion as the curvature









119894119896(119872)

where Det|119892119894119896| = 0 and 119892




119894119896119889119909119894



119894119896= 119892119896119894


1198891199042

= 119892119894119896119889119909119894

119889119909119896

(4)




119904 = int

119887

119886

radic119892119894119896119889119909119894119889119909119896

= int

119887

119886

radic119892119894119896(1199091(119905) 119909

119899(119905))

119889119909119894

(119905)

119889119905

119889119909119896

(119905)

119889119905

119889119905

(5)


119895119896(119872) that is based on 119899-


119894



119894 to another 1199091198941015840

in the form of

Γ1198941015840

11989510158401198961015840 = Γ119894

119895119896

1205971199091198941015840

120597119909119894

120597119909119895

1205971199091198951015840

120597119909119896

1205971199091198961015840+

1205972

119909119894

1205971199091198951015840

1205971199091198961015840

1205971199091198941015840

120597119909119894 (6)




119894


119889119860119894

= minusΓ119894

119895119896119860119895

119889119909119896

(7)



119894

119895119896(119872) An inter-




119889 (119892119894119896119860119894

119861119896

) = 0 (8)



119895119896= Γ119894







119878119894

119895119896equiv Γ119894

119895119896minus Γ119894

119896119895 (9)





If demand 119889(119892119894119896119860119894

119861119896



is uniquelydefined



We rewrite 119889(119892119894119896119860119894

119861119896

) = 0 as (119892119894119896119897

minus 119892119898119896Γ119898

119894119897minus

119892119894119898Γ119898

119896119897)119860119894

119861119896

119889119909119897


119901119897119860119901

119889119909119897




equiv

(120597120597119909119897



119892119894119896119897

= 119892119898119896Γ119898

119894119897+ 119892119894119898Γ119898

119896119897

119892119897119894119896

= 119892119898119894Γ119898

119897119896+ 119892119897119898Γ119898

119894119896

119892119896119897119894

= 119892119898119897Γ119898

119896119894+ 119892119896119898Γ119898

119897119894

(10)


119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

= 119892119898119896119878119898

119894119897+ 119892119898119897119878119898

119894119896+ 119892119894119898Γ119898

119896119897+ 119892119898119894Γ119898

119897119896 (11)

where 119878119898119894119897= Γ119898

119894119897minus Γ119898


119892119894119898(Γ119898

119896119897+ Γ119898

119897119896)

= 119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

+ 119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894

Γ119901

119896119897+ Γ119901

119897119896

= 119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

+ 119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

(12)


119896119897minus

Γ119901

119897119896= 119878119901


Γ119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

+ 119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

+

1

2

119878119901

119896119897

(13)


119875119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894) (14)


119871119901

119896119897equiv

1

2

119878119901

119896119897+

1

2

119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) (15)


119901



119898

119894119896


119875119901


119894119896


119894119896and the tensor 119878119898

119896119897 namely

Γ119901

119896119897= 119875119901

119896119897+ 119871119901

119896119897 (16)



symmetric (12)119892119901119894(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) and torsion (12)119878119901

119896119897


Γ119901

119901119897=

1

2

119892119894119901119897119892119894119901

=

1

radic119892

120597radic119892

120597119909119897

where 119892 = det 1003816100381610038161003816119892119894119896

1003816100381610038161003816 (17)


119889119860119894

= minusΓ119894

119901119897119860119901

119889119909119897

(18)




119906119894119897equiv 119906119894119897minus Γ119896

119894119897119906119896

119906119894

119897equiv 119906119894

119897+ Γ119894

119896119897119906119896

(19)



119894119897119906119896 (20)



119885119896

= 119878119896

119894119895119860119894

119861119895

1205912

(21)


119895



119906119894119897119896

minus 119906119894119896119897

= 119877119901

119896119897119894119906119901+ 119878119902

119896119897119906119894119902 (22)

where we identified

119877119901

119896119897119894equiv Γ119901

119894119896119897minus Γ119901

119894119897119896+ Γ119901

119902119897Γ119902

119894119896minus Γ119901

119902119896Γ119902

119894119897 (23)

119877119901


119906119894

119897119896minus 119906119894

119896119897= minus119877119894

119896119897119901119906119901

+ 119878119902

119896119897119906119894

119902 (24)


119863119860119894

equivsim 119889119860119894

+ Γ119894

119895119896119860119895

119889119909119896

= (119860119894

119896+ Γ119894

119895119896119860119895

) 119889119909119896

(25)



119863119863119860119894

minus 119863119863119860119894

= minus119877119894

119896119897119901119860119901 119889119909119896

119889119909119897

(26)


119889119909119897



119877119901

119894119896119897= Γ119901

119897119894119896minus Γ119901

119897119896119894+ Γ119901

119902119896Γ119902

119897119894minus Γ119901

119902119894Γ119902

119897119896 (27)


119877119901

119894119896119897= 119875119901

119897119894119896+ 119871119901

119897119894119896minus 119875119901

119897119896119894+ 119871119901

119897119896119894+ (119875119901

119902119896+ 119871119901

119902119896) (119875119901

119897119894+ 119871119902

119897119894)

minus (119875119901

119902119894+ 119871119901

119902119894) (119875119901

119897119896+ 119871119902

119897119896)

= 119875119901

119897119894119896minus 119875119901

119897119896119894+ 119875119901

119902119896119875119902

119897119894minus 119875119901

119902119894119875119902

119897119896+ 119871119901

119897119894119896minus 119871119901

119897119896119894

+ 119875119901

119902119896119871119902

119897119894+ 119875119902

119897119894119871119901

119902119896minus 119875119901

119902119894119871119902

119897119896minus 119875119902

119897119896119871119901

119902119894

+ 119871119901

119902119896119871119902

119897119894minus 119871119901

119902119894119871119902

119897119896

(28)


119875119901

119894119896119897equiv 119875119901

119897119894119896minus 119875119901

119897119896119894+ 119875119901

119902119896119875119902

119897119894+ 119875119901

119902119894119875119902

119897119896(29)


119885119901

119894119896119897equiv 119871119901

119902119896119871119902

119897119894minus 119871119901

119902119894119871119902

119897119896(30)

is tensor and

119879119901

119894119896119897equiv 119871119901

119897119894119896minus 119871119901

119897119896119894+ 119875119901

119902119896119871119902

119897119894+ 119875119902

119897119894119871119901

119902119896minus 119875119901

119902119894119871119902

119897119896minus 119875119902

119897119896119871119901

119902119894(31)




119879119901

119894119896119897= 119871119901

119897119894119896minus 119871119901

119897119896119894minus 119871119902

119897119894Γ119901

119902119896+ 119871119901

119902119894Γ119902

119897119896+ 119871119901

119897119902Γ119902

119894119896+ 119871119902

119897119896Γ119901

119902119894

minus 119871119901

119902119896Γ119902

119897119894minus 119871119901

119897119902Γ119902

119896119894+ 119875119901

119902119896119871119902

119897119894+ 119875119902

119897119894119871119902

119902119896

minus 119875119901

119902119894119871119902

119897119896minus 119875119902

119897119896119871119901

119902119894

= 119871119901

119897119894119896minus 119871119901

119897119896119894minus 119871119902

119897119894119871119901

119902119896+ 119871119901

119902119894119871119902

119897119896+ 119871119902

119897119896119871119901

119902119894

minus 119871119901

119902119896119871119902

119897119894+ 119871119901

119897119902119878119902

119894119896

(32)



119872119901

119894119896119897equiv 119879119901

119894119896119897+ 119885119901

119894119896119897 (33)

then we obtain

119872119901

119894119896119897= 119871119901

119897119894119896minus 119871119901

119897119896119894+ 119871119901

119897119902119878119902

119894119896+ 119871119901

119902119894119871119902

119897119896minus 119871119901

119902119896119871119902

119897119894 (34)


119877119901

119894119896119897= 119875119901

119894119896119897+119872119901

119894119896119897 (35)




119895119896


119877119901

119894119896119897+ 119877119901

119896119897119894+ 119877119901

119897119894119896= 119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896+ Γ119901

119902119896119878119902

119897119894+ Γ119901

119902119896119878119902

119894119896+ Γ119901

119902119894119878119902

119896119897

= 119878119901

119894119896119897+ Γ119902

119894119897119878119901

119902119896+ Γ119902

119896119897119878119901

119894119902+ 119878119901

119896119897119894+ Γ119902

119896119894119878119901

119902119897+ Γ119902

119897119894119878119901

119896119902

+ 119878119901

119897119894119896+ Γ119902

119897119896119878119901

119902119894+ Γ119902

119894119896119878119901

119897119902

= 119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896

+ 119878119901

119897119902119878119902

119894119896+ 119878119901

119896119902119878119902

119897119894+ 119878119901

119894119902119878119902

119896119897

(36)


119878119894

119895119901119878119901

119896119894+ 119878119894

119896119901119878119901

119894119895= 0

119878119894

119894119901119878119901

119895119896= 0

(37)


119878119894

119895119901119878119901

119896119894+ 119878119894

119896119901119878119901

119894119895+ 119878119894

119894119901119878119901

119895119896= 0 (38)




1198751199011015840

11989610158401198971015840= 119875119901

119896119897

1205971199091199011015840

120597119909119901

120597119909119896

1205971199091198961015840

120597119909119897

1205971199091198971015840+

1205972

119909119901

1205971199091198961015840

1205971199091198971015840

1205971199091199011015840

120597119909119901 (39)


119895119896equiv Γ119894

119895119896minus Γ119894

119896119895)




Value

119872119901

119896119897equiv

1

2

119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) (40)




1198892

119909119896

1198891205912= minusΓ119896

119894119895

119889119909119894

119889120591

119889119909119895

119889120591

(41)


119896119897+119872119901

119896119897 In (6) 120591


1198892

119909119896

1198891199042

= minusΓ119896

119894119895

119889119909119894

119889119904

119889119909119895

119889119904

(42)




119875119901

119896119897+119872119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894)

+

1

2

119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

(43)



1198892

119909119896

1198891199042

= minusΓ119896

119894119895

119889119909119894

119889119904

119889119909119895

119889119904

(44)



= 119909119894

(119905) 119905 isin [1199051 1199052] We


120575119878 = int

1199052

1199051

120575radic119892119894119895

119889119909119894

119889119905

119889119909119895

119889119905

119889119905

= int

1199052

1199051

120575 (119892119894119895(119889119909119894

119889119905) (119889119909119895

119889119905))

2radic119892119894119895(119889119909119894119889119905) (119889119909

119895119889119905)

119889119905

120575 (119892119894119895

119889119909119894

119889119905

119889119909119895

119889119905

) = 119892119894119895119863

119889119909119894

119889119905

119889119909119895

119889119905

+ 119892119894119895

119889119909119894

119889119905

119863

119889119909119895

119889119905

= 2119892119894119895

119889119909119894

119889119905

119863

119889119909119895

119889119905

119863

119889119909119895

119889119905

= 120575

119889119909119895

119889119905

+ Γ119895

119901119896

119889119909119901

119889119905

120575119909119896

119863

120575119909119895

119889119905

=

119889

119889119905

120575119909119895

+ Γ119895

119896119901

119889119909119896

119889119905

120575119909119901

119863

119889119909119895

119889119905

= 119863

120575119909119895

119889119905

+ 119878119895

119901119896

119889119909119901

119889119905

120575119909119896

(45)


120575(119892119894119895

119889119909119894

119889119905

119889119909119895

119889119905

) = 2119892119894119895

119889119909119894

119889119905

(119863

120575119909119895

119889119905

+ 119878119895

119901119896

119889119909119901

119889119905

120575119909119896

)

120575119904 = int

1199052

1199051

119892119894119895

119889119909119894

119889119904

119863120575119909119895

+ int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119904

119889119909119901

120575119909119896

= int

1199052

1199051

119863(119892119894119895

119889119909119894

119889119904

120575119909119895

)

minus int

1199052

1199051

119892119894119895119863

119889119909119894

119889119904

120575119909119895

+ int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119904

119889119909119901

120575119909119896

(46)


120575119904 = int

1199052

1199051

(119892119894119895119878119895

119901119896

119889119909119894

119889119905

119889119909119901

120575119909119896

minus 119892119894119895119863

119889119909119894

119889119905

120575119909119895

) (47)


int

1199052

1199051

(119892119894119895119878119895

119901119896

119889119909119894

119889119904

119889119909119901

120575119909119896

minus 119892119894119895119863

119889119909119894

119889119904

120575119909119895

) = 0 (48)


119892119894119896119878119896

119901119895

119889119909119894

119889119904

119889119909119901

minus 119892119894119895119863

119889119909119894

119889119904

= 0 (49)


119902

119892119895119902

119892119894119896119878119896

119901119895120585119894

119889119909119901


120575119904 = int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119905

119889119909119901

120575119909119896

(50)


119894119896and 119878

119895





119895



int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119905

119889119909119901

120575119909119896

(51)


119894119895


119896

= 119878119896

119894119895119860119894

119861119895


119894

120591 and 119861119895

120591


119901119902119878119901

119894119895119860119894

119861119895

119878119902

119896119897119860119896

119861119897

1205914


119894119896) with







119894119895


1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (52)

that is

Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (53)


Γ119896

119894119895=

1

2

119892119896119901

(119892119894119901119895119897

+ 119892119895119901119894

minus 119892119894119895119901

+ 119892119894119898119878119898

119895119901+ 119892119895119898119878119898

119894119901)

+

1

2

119878119896

119894119895

(54)



Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (55)



Γ119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

+ 119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

+

1

2

119878119901

119896119897

(56)


119875119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894) (57)


119901

119896119897+ 119872119901

119896119897 here 119872119896



119861119894

= 119886119860119894

(58)


119889119860119896

= minus (119875119896

119894119895+119872119896

119894119895)119860119894

119889119909119895

119889119861119896

= minus119875119896

119894119895119861119894

119889119909119895

(59)

Then we have

119860119896

119889119886 + 119886119889119860119896

= minus119875119896

119894119895119886119860119894

119889119909119895

(60)

119860119896

119889119886

119886

= 119872119896

119894119895119860119894

119889119909119895

(61)



119896119897+119872119901

119896119897


119889 ln 119886119889120591

119860119896

= 119872119896

119894119895119860119894

119860119895

(62)



119860119897

119872119896

119894119895119860119894

119860119895

minus 119860119896

119872119897

119894119895119860119894

119860119895

= 0 (63)

or

120575119897

119898119872119896

119894119895119860119894

119860119895

119860119898

minus 120575119896

119898119872119897

119894119895119860119894

119860119895

119860119898

= 0 (64)




120575119897

119898119872119896

119894119895minus 120575119896

119898119872119897

119894119895+ 120575119897

119894119872119896

119895119898minus 120575119896

119894119872119897

119895119898+ 120575119897

119895119872119896

119898119894minus 120575119896

119895119872119897

119898119894

= 0

(65)


have

119872119896

119894119895=

1

119899 + 1

(120575119896

119894119872119897

119895119897+ 120575119896

119895119872119897

119894119897) (66)



119894119895


119894119895 then let 119872119901

119896119897equiv

(12)119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) then119872119897

119896119897equiv (12)119878

119897

119896119897

We obtained

Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (67)



119894119895


119894119895=

(12)(1(119899 + 1))(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897



119896

119894119895119860119894

119889119909119895



119894119895







119894119895= 0 Assume that 119894 119872 sub 119875


119909rarr

119879119909119875|119872

rarr 119879perp



perp



119883119884 = 119879(nabla

1015840

119883119884) + perp (nabla

1015840

119883119884) Let it be

nabla119883119884 = 119879(nabla

1015840

119883119884) 120572(119883 119884) = perp (nabla




⟨1198771015840

(119883 119884)119885119882⟩ = ⟨119877 (119883 119884)119885119882⟩

+ ⟨120572 (119883 119885) 120572 (119884119882)⟩

minus ⟨120572 (119884 119885) 120572 (119883119882)⟩

(68)



(nabla10158401015840

119883120572) (119884 119885) = 119863

119883(120572 (119884 119885)) minus 120572 (nabla

119883119884119885)

minus 120572 (119884 nabla119883119885)

(69)

Codazzi equation is

perp ⟨1198771015840

(119883 119884)119885⟩ = (nabla10158401015840

119883120572) (119884 119885) minus (nabla

10158401015840

119884120572) (119883 119885) (70)





119909119894

= 119909119894

(1199101

119910119899minus1

) (71)


119886120572120573

= 119892119894119895

120597119909119894

120597119910120572

120597119909119895

120597119910120573

(72)


120572120573of



we assume that 119866120572120573120574



120572120573


119894119895and

119878119896

119894119895 we have

119866120572

120573120574

=

1

2

(119886120572120578

(119886120573120578120574

+ 119886120574120578120573

+ 119886120573120574120578

+ 119886120573120583119879120583

120574120578+ 119886120574120583119879120583

120573120578) + 119879120572

120574120573)

(73)


119866120572



1015840

by the formula

119866120572

120573120574= 1198661205721015840

12057310158401205741015840

120597119906120572

1205971199061205721015840

1205971199061205731015840

120597119906120573

1205971199061205741015840

120597119906120574+

120597119906120572

1205971199061205721015840

1205972

1199061205721015840

120597119906120573120597119906120574

(74)




119866120572

120573120574

120597119909119896

120597119906120572= Γ119896

119894119895

120597119909119894

120597119906120573

120597119909119895

120597119906120574+

1205972

119909119896

120597119906120573120597119906120574

(75)

We obtain

1

2

(119886120572120578

(119886120573120578120574

+ 119886120574120578120573

+ 119886120573120574120578

+ 119886120573120583119879120583

120574120578+ 119886120574120583119879120583

120573120578) + 119879120572

120574120573)

120597119909119896

120597119906120572

=

1

2

(119892119896119899

(119892119899119894119895

+ 119892119899119895119894

minus 119892119894119895119899

+ 119892119894119898119878119898

119895119899+ 119892119895119898119878119898

119894119899) + 119878119896

119894119895)

times

120597119909119894

120597119906120573

120597119909119895

120597119906120574+

1205972

119909119896

120597119906120573120597119906120574

(76)


119879120574

120572120573= 119886120574120578

119892119901119902119878119901

119894119895

120597119909119894

120597119910120572

120597119909119895

120597119910120573

120597119909119902

120597119910120578 (77)

using tensors 119886120572120573

and 119879120574












119860119894120572

119895120573120574= 119860119894120572

119895120573120574+ Γ119894

119901119896119860119901120572

119895120573

120597119909119896

120597119910120574minus Γ119901

119895119902119860119894120572

119901120573

120597119909119902

120597119910120574+ 119866120572

120578120574119860119894120578

119895120573

minus 119866120578

120573120574119860119894120572

119895120578

(78)


119906119894120572120573

minus 119906119894120573120572

= 119877119901

119896119897119894119906119901

120597119909119897

120597119910120572

120597119909119896

120597119910120573

+ 119878119902

119896119897119906119894119902

120597119909119897

120597119910120572

120597119909119896

120597119910120573

119906120574120572120573

minus 119906120574120573120572

= 119877120578

120573120572120574119906120578+ 119879120578

120573120572119906120574120578

(79)

where 119877120578



120573120574


120585119894

120572=

120597119909119894

120597119910120572 (80)



120585119894

1 120585

119894

119899minus1 ]119894 (81)

where 120585119894

1 120585

119894




120572

120585119894

120572120574= 120585119894

120572120574+ Γ119894

119901119902120585119901

120572

120597119909119902

120597119910120574minus 119866120578

120572120574120585119894

120578 (82)


120585119894

120572120574minus 120585119894

120574120572= 119878119894

119901119902120585119901

120572120585119902

120574+ 119879120578

120574120572120585119894

120578 (83)


0 = 119886120572120573120574

= (119892119894119895120585119894

120572120585119895

120573)

120574

= 119892119894119895120585119894

120572120574120585119895

120573+ 119892119894119895120585119894

120572120585119895

120573120574 (84)

we obtain

119892119894119895120585119894

120572120574120585119895

120573= 0 (85)


120585119894

120573120572= 120587120572120573]119894 (86)

Remark 11 120587120572120573



120574) the formula

119892119894119895]119894120574120585119894

120572= minus120587120574120572 (87)


]119894120574= minus119886120578120583

120587120583120574120585119894

120578 (88)



Further we obtain

120585119894

120573120594120582minus 120585119894

120573120582120594= minus119877119894

119896119897119901120585119896

120582120585119897

120594120585119901

120573+ 119877120590

120582120594120573120585119894

120590+ 119879120590

120582120594120585119894

120573120590

= (120587120594120573120582

minus 120587120582120573120594

) ]119894

minus (120587120594120573120587120578120582119886120578120590

minus 120587120582120573120587120578120594119886120578120590

) 120585119894

120590

(89)


120572 we have

119877120572120582120594120573

= 119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573120585119894

120572minus (120587120594120573120587120572120582

minus 120587120582120573120587120572120594) (90)


]119894120594120582

minus ]119894120582120594

= minus119877119894

119896119897119901120585119896

120582120585119897

120594]119901 + 119879

120590

120582120594]119894120590

= (120587120578120582120594

119886120578120590

minus 120587120578120594120582

119886120578120590

) 120585119894

120590

(91)


minus119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573]119894 + 119879

120590

120582120594120587120590120573

= 120587120594120573120582

minus 120587120582120573120594

(92)



minus119877119894119896119897119901

120585119896

120582120585119897

120594]119901120585119894120572+ 119879120590

120582120594120587120572120590

= 120587120572120582120594

minus 120587120572120594120582

(93)










119878119894

119895119896119901119902minus 119878119894

119895119896119902119901= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

+ 119878119905

119902119901119878119894

119895119896119905

(94)

or

119878119894

119895119896119901119902minus 119878119894

119895119896119902119901minus 119878119905

119902119901119878119894

119895119896119905

= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

(95)


119878119894

119895119896119901119894minus 119878119894

119895119896119894119901minus 119878119905

119894119901119878119894

119895119896119905

= (119878119894

119895119896119901minus 119878119894

119902119901119878119902

119895119896)

119894

minus 119878119894

119895119896119894119901minus 119878119894

119901119902119894119878119902

119895119896

(96)


(119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896)119894

minus 119892119896119901

119892119895119904

119878119894

119904119896119894119901minus 119892119896119901

119892119895119904

119878119894

119901119902119894119878119902

119904119896

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905

minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(97)


119867119895119894

= 119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896

119865119895119901

= 119892119896119901

119892119895119904

119878119894

119904119896119894

(98)


119867119895119894

119894minus 119865119895119894

119894minus 119892119896119901

119892119895119904

119878119902

119904119896119865119901119902

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(99)

where 119865119901119902

= 119878119894

119901119902119894


119894119896119897+ 119877119901

119896119897119894+ 119877119901

119897119894119896= 0 hence

119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896+ 119878119901

119897119902119878119902

119894119896+ 119878119901

119896119902119878119902

119897119894+ 119878119901

119894119902119878119902

119896119897= 0 (100)


119878119901

119894119896119901+ 119878119901

119896119901119894+ 119878119901

119901119894119896= 0 (101)




119894119895119878119902


expression

119878119901

119894119895119901= 120593119894119895minus 120593119895119894 (102)

Next if we put 119878119901119894119895119901


=



119894 System (99)

takes the form

119867119895119894

119894= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

119865119894119895

= 0

(103)


119862119894119895119896

= 119892119901119895

119892119902119896

119878119894

119901119902+ 119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896

119901119902 (104)


119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+ 119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904

minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

(105)



119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

=

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

(106)

hence

119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902) (107)


119862119894119896119895

119894= minus119862119894119895119896

119894= minus (119862

119894119895119896

119894+ Γ119895

119901119894119862119894119901119896

+ Γ119896

119901119894119862119894119895119901

+ Γ119894

119901119894119862119901119896119895

) (108)


= 119892119901119895

119892119902119896

119878119894

119901119902+

119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896


Γ119895

119901119894119862119894119901119896

= Γ119895

119894119901119862119894119901119896

=

1

2

(Γ119895

119894119901119862119894119901119896

+ Γ119895

119901119894119862119901119894119896

) =

1

2

119878119895

119894119901119862119901119896119894

= minus

1

2

119878119895

119901119894119862119896119901119894

(109)

similarly we obtain

Γ119896

119901119894119862119894119895119901

= Γ119896

119894119901119862119901119895119894

=

1

2

119862119895119901119894

(Γ119896

119894119901minus Γ119896

119901119894) =

1

2

119878119896

119894119901119862119895119901119894

=

1

2

119878119896

119901119902119862119895119902119901

(110)

Then we write

119862119894119896119895

119894= minus119862119894119895119896

119894

= minus119862119894119895119896

119894minus

1

2

119878119895

119901119902119862119896119901119902

+

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894+

1

2

119878119895

119901119902119862119896119901119902

minus

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

(111)



119901

119901119897=

(12)119892119894119901119897119892119894119901

= (1radic119892)(120597radic119892120597119909119897

) and Γ119901

119897119901= Γ119901

119901119897+ 119878119901

119897119901and

obtain

Γ119901

119901119897=

1

radicminus119892


120597119909119897

+ (ln120595)119897= (ln (120595radicminus119892))

119897 (112)

Then we obtain

119867119895119896

minus 119867119896119895

minus 119865119895119896

= minus119862119894119895119896


119894119862119894119896119895

(113)


120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


119894+ (ln (120595radicminus119892))

119894119862119894119896119895

)

120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


)119894

(114)


(120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

))119896

= 0 (115)







119894

119895119896 we



119894

119895119896and 119892

119894119896


119877119894119895= Γ119901

119894119901119895minus Γ119901

119894119895119901+ Γ119901

119902119895Γ119902

119894119901minus Γ119901

119902119901Γ119902

119894119895 (116)


119894119897119878119897

119896119895)119892119894119896


Γ119894

119895119896minus Γ119894


int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881 (117)




119877119894119896+ 119878119895

119894119897119878119897

119896119895= 0 (118)


119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901

minus 120575119895

119896(119892119894119901

119901minus

1

2

119892119894119901

119892119898119899119892119898119899

119901+ 119892119901119902

Γ119894

119901119902) minus 119892119894119895

Γ119901

119896119901

+ 2 (119892119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(119)



minus3119892119894119895

119895+

3

2

119892119894119895

119892119901119902119892119901119902

119895minus 3119892119901119902

Γ119894

119901119902+ 119892119894119901

119878119895

119901119895= 0 (120)

Therefore we have

119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901minus 119892119894119895

Γ119901

119896119901

minus

1

3

120575119895

119896119892119894119901

119878119902

119901119902+ 2 (119892

119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(121)


minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901+ 2 (119892

119901119895119878119901

119894119896+ 119892119894119901119878119901

119896119895) = 0

(122)



5 (119892119894119901119878119901

119896119895+ 119892119895119901119878119901

119894119896) +

1

3

(119892119895119896119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0 (123)

Further we obtained

minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901minus

2

15

(119892119896119895119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0

(124)


0 then from (123) we have 119878119894




119898+

119882119892) = 0 where 119882

119898and 119882




120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int (119877119894119896radicminus119892120575119892

119894119896

+ 119877119894119896119892119894119896

120575radicminus119892 + 119892119894119896

radicminus119892120575119877119894119896

+ 119878119895

119894119897119878119897

119896119895radicminus119892120575119892

119894119896

+ 119878119895

119894119897119878119897

119896119895119892119894119896

120575radicminus119892) 119889119881

119877119894119896119892119894119896


1

2

119877119901119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(125)

Similarly we obtain

119878119895

119894119897119878119897

119896119895119892119894119896


1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(126)



119894119896

(120575Γ119897

119896119894)119897minus 119892119894119896

(120575Γ119897

119896119897)119894= (119892119894119896

120575Γ119897

119896119894minus 119892119894119897

120575Γ119901

119897119901)119897



119894119896= 119892119894119896

120575(Γ119901

119902119901Γ119902

119896119894minus

Γ119901

119902119894Γ119902119897



119892119894119896

120575119877119894119896= (119878119901

119894119901119878119902

119902119896minus 119878119901

119894119902119878119902

119901119896minus 119892119896119901119892119902119905

119878119898

119894119905119878119901

119902119898) 120575119892119894119896

(127)

Thus we have

120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(128)


120575119882119892

= 1198701int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= 1198701int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(129)


1198883


120575119882119892= 1198702int119879119894119896radicminus119892120575119892

119894119896

119889119881 (130)


2usually


119892+120575119882119898=

0 we find relations

int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902


119894119896

119889119881 = 0

(131)



119877119894119896minus

1

2

119892119894119896119877 + 119878119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902+ 119878119895

119894119897119878119897

119896119895

minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896= 119870119879119894119896

(132)


2

7 Conclusions







References




























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics











119894119896(119872)

where Det|119892119894119896| = 0 and 119892




119894119896119889119909119894



119894119896= 119892119896119894


1198891199042

= 119892119894119896119889119909119894

119889119909119896

(4)




119904 = int

119887

119886

radic119892119894119896119889119909119894119889119909119896

= int

119887

119886

radic119892119894119896(1199091(119905) 119909

119899(119905))

119889119909119894

(119905)

119889119905

119889119909119896

(119905)

119889119905

119889119905

(5)


119895119896(119872) that is based on 119899-


119894



119894 to another 1199091198941015840

in the form of

Γ1198941015840

11989510158401198961015840 = Γ119894

119895119896

1205971199091198941015840

120597119909119894

120597119909119895

1205971199091198951015840

120597119909119896

1205971199091198961015840+

1205972

119909119894

1205971199091198951015840

1205971199091198961015840

1205971199091198941015840

120597119909119894 (6)




119894


119889119860119894

= minusΓ119894

119895119896119860119895

119889119909119896

(7)



119894

119895119896(119872) An inter-




119889 (119892119894119896119860119894

119861119896

) = 0 (8)



119895119896= Γ119894







119878119894

119895119896equiv Γ119894

119895119896minus Γ119894

119896119895 (9)





If demand 119889(119892119894119896119860119894

119861119896



is uniquelydefined



We rewrite 119889(119892119894119896119860119894

119861119896

) = 0 as (119892119894119896119897

minus 119892119898119896Γ119898

119894119897minus

119892119894119898Γ119898

119896119897)119860119894

119861119896

119889119909119897


119901119897119860119901

119889119909119897




equiv

(120597120597119909119897



119892119894119896119897

= 119892119898119896Γ119898

119894119897+ 119892119894119898Γ119898

119896119897

119892119897119894119896

= 119892119898119894Γ119898

119897119896+ 119892119897119898Γ119898

119894119896

119892119896119897119894

= 119892119898119897Γ119898

119896119894+ 119892119896119898Γ119898

119897119894

(10)


119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

= 119892119898119896119878119898

119894119897+ 119892119898119897119878119898

119894119896+ 119892119894119898Γ119898

119896119897+ 119892119898119894Γ119898

119897119896 (11)

where 119878119898119894119897= Γ119898

119894119897minus Γ119898


119892119894119898(Γ119898

119896119897+ Γ119898

119897119896)

= 119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

+ 119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894

Γ119901

119896119897+ Γ119901

119897119896

= 119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

+ 119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

(12)


119896119897minus

Γ119901

119897119896= 119878119901


Γ119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

+ 119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

+

1

2

119878119901

119896119897

(13)


119875119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894) (14)


119871119901

119896119897equiv

1

2

119878119901

119896119897+

1

2

119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) (15)


119901



119898

119894119896


119875119901


119894119896


119894119896and the tensor 119878119898

119896119897 namely

Γ119901

119896119897= 119875119901

119896119897+ 119871119901

119896119897 (16)



symmetric (12)119892119901119894(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) and torsion (12)119878119901

119896119897


Γ119901

119901119897=

1

2

119892119894119901119897119892119894119901

=

1

radic119892

120597radic119892

120597119909119897

where 119892 = det 1003816100381610038161003816119892119894119896

1003816100381610038161003816 (17)


119889119860119894

= minusΓ119894

119901119897119860119901

119889119909119897

(18)




119906119894119897equiv 119906119894119897minus Γ119896

119894119897119906119896

119906119894

119897equiv 119906119894

119897+ Γ119894

119896119897119906119896

(19)



119894119897119906119896 (20)



119885119896

= 119878119896

119894119895119860119894

119861119895

1205912

(21)


119895



119906119894119897119896

minus 119906119894119896119897

= 119877119901

119896119897119894119906119901+ 119878119902

119896119897119906119894119902 (22)

where we identified

119877119901

119896119897119894equiv Γ119901

119894119896119897minus Γ119901

119894119897119896+ Γ119901

119902119897Γ119902

119894119896minus Γ119901

119902119896Γ119902

119894119897 (23)

119877119901


119906119894

119897119896minus 119906119894

119896119897= minus119877119894

119896119897119901119906119901

+ 119878119902

119896119897119906119894

119902 (24)


119863119860119894

equivsim 119889119860119894

+ Γ119894

119895119896119860119895

119889119909119896

= (119860119894

119896+ Γ119894

119895119896119860119895

) 119889119909119896

(25)



119863119863119860119894

minus 119863119863119860119894

= minus119877119894

119896119897119901119860119901 119889119909119896

119889119909119897

(26)


119889119909119897



119877119901

119894119896119897= Γ119901

119897119894119896minus Γ119901

119897119896119894+ Γ119901

119902119896Γ119902

119897119894minus Γ119901

119902119894Γ119902

119897119896 (27)


119877119901

119894119896119897= 119875119901

119897119894119896+ 119871119901

119897119894119896minus 119875119901

119897119896119894+ 119871119901

119897119896119894+ (119875119901

119902119896+ 119871119901

119902119896) (119875119901

119897119894+ 119871119902

119897119894)

minus (119875119901

119902119894+ 119871119901

119902119894) (119875119901

119897119896+ 119871119902

119897119896)

= 119875119901

119897119894119896minus 119875119901

119897119896119894+ 119875119901

119902119896119875119902

119897119894minus 119875119901

119902119894119875119902

119897119896+ 119871119901

119897119894119896minus 119871119901

119897119896119894

+ 119875119901

119902119896119871119902

119897119894+ 119875119902

119897119894119871119901

119902119896minus 119875119901

119902119894119871119902

119897119896minus 119875119902

119897119896119871119901

119902119894

+ 119871119901

119902119896119871119902

119897119894minus 119871119901

119902119894119871119902

119897119896

(28)


119875119901

119894119896119897equiv 119875119901

119897119894119896minus 119875119901

119897119896119894+ 119875119901

119902119896119875119902

119897119894+ 119875119901

119902119894119875119902

119897119896(29)


119885119901

119894119896119897equiv 119871119901

119902119896119871119902

119897119894minus 119871119901

119902119894119871119902

119897119896(30)

is tensor and

119879119901

119894119896119897equiv 119871119901

119897119894119896minus 119871119901

119897119896119894+ 119875119901

119902119896119871119902

119897119894+ 119875119902

119897119894119871119901

119902119896minus 119875119901

119902119894119871119902

119897119896minus 119875119902

119897119896119871119901

119902119894(31)




119879119901

119894119896119897= 119871119901

119897119894119896minus 119871119901

119897119896119894minus 119871119902

119897119894Γ119901

119902119896+ 119871119901

119902119894Γ119902

119897119896+ 119871119901

119897119902Γ119902

119894119896+ 119871119902

119897119896Γ119901

119902119894

minus 119871119901

119902119896Γ119902

119897119894minus 119871119901

119897119902Γ119902

119896119894+ 119875119901

119902119896119871119902

119897119894+ 119875119902

119897119894119871119902

119902119896

minus 119875119901

119902119894119871119902

119897119896minus 119875119902

119897119896119871119901

119902119894

= 119871119901

119897119894119896minus 119871119901

119897119896119894minus 119871119902

119897119894119871119901

119902119896+ 119871119901

119902119894119871119902

119897119896+ 119871119902

119897119896119871119901

119902119894

minus 119871119901

119902119896119871119902

119897119894+ 119871119901

119897119902119878119902

119894119896

(32)



119872119901

119894119896119897equiv 119879119901

119894119896119897+ 119885119901

119894119896119897 (33)

then we obtain

119872119901

119894119896119897= 119871119901

119897119894119896minus 119871119901

119897119896119894+ 119871119901

119897119902119878119902

119894119896+ 119871119901

119902119894119871119902

119897119896minus 119871119901

119902119896119871119902

119897119894 (34)


119877119901

119894119896119897= 119875119901

119894119896119897+119872119901

119894119896119897 (35)




119895119896


119877119901

119894119896119897+ 119877119901

119896119897119894+ 119877119901

119897119894119896= 119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896+ Γ119901

119902119896119878119902

119897119894+ Γ119901

119902119896119878119902

119894119896+ Γ119901

119902119894119878119902

119896119897

= 119878119901

119894119896119897+ Γ119902

119894119897119878119901

119902119896+ Γ119902

119896119897119878119901

119894119902+ 119878119901

119896119897119894+ Γ119902

119896119894119878119901

119902119897+ Γ119902

119897119894119878119901

119896119902

+ 119878119901

119897119894119896+ Γ119902

119897119896119878119901

119902119894+ Γ119902

119894119896119878119901

119897119902

= 119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896

+ 119878119901

119897119902119878119902

119894119896+ 119878119901

119896119902119878119902

119897119894+ 119878119901

119894119902119878119902

119896119897

(36)


119878119894

119895119901119878119901

119896119894+ 119878119894

119896119901119878119901

119894119895= 0

119878119894

119894119901119878119901

119895119896= 0

(37)


119878119894

119895119901119878119901

119896119894+ 119878119894

119896119901119878119901

119894119895+ 119878119894

119894119901119878119901

119895119896= 0 (38)




1198751199011015840

11989610158401198971015840= 119875119901

119896119897

1205971199091199011015840

120597119909119901

120597119909119896

1205971199091198961015840

120597119909119897

1205971199091198971015840+

1205972

119909119901

1205971199091198961015840

1205971199091198971015840

1205971199091199011015840

120597119909119901 (39)


119895119896equiv Γ119894

119895119896minus Γ119894

119896119895)




Value

119872119901

119896119897equiv

1

2

119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) (40)




1198892

119909119896

1198891205912= minusΓ119896

119894119895

119889119909119894

119889120591

119889119909119895

119889120591

(41)


119896119897+119872119901

119896119897 In (6) 120591


1198892

119909119896

1198891199042

= minusΓ119896

119894119895

119889119909119894

119889119904

119889119909119895

119889119904

(42)




119875119901

119896119897+119872119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894)

+

1

2

119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

(43)



1198892

119909119896

1198891199042

= minusΓ119896

119894119895

119889119909119894

119889119904

119889119909119895

119889119904

(44)



= 119909119894

(119905) 119905 isin [1199051 1199052] We


120575119878 = int

1199052

1199051

120575radic119892119894119895

119889119909119894

119889119905

119889119909119895

119889119905

119889119905

= int

1199052

1199051

120575 (119892119894119895(119889119909119894

119889119905) (119889119909119895

119889119905))

2radic119892119894119895(119889119909119894119889119905) (119889119909

119895119889119905)

119889119905

120575 (119892119894119895

119889119909119894

119889119905

119889119909119895

119889119905

) = 119892119894119895119863

119889119909119894

119889119905

119889119909119895

119889119905

+ 119892119894119895

119889119909119894

119889119905

119863

119889119909119895

119889119905

= 2119892119894119895

119889119909119894

119889119905

119863

119889119909119895

119889119905

119863

119889119909119895

119889119905

= 120575

119889119909119895

119889119905

+ Γ119895

119901119896

119889119909119901

119889119905

120575119909119896

119863

120575119909119895

119889119905

=

119889

119889119905

120575119909119895

+ Γ119895

119896119901

119889119909119896

119889119905

120575119909119901

119863

119889119909119895

119889119905

= 119863

120575119909119895

119889119905

+ 119878119895

119901119896

119889119909119901

119889119905

120575119909119896

(45)


120575(119892119894119895

119889119909119894

119889119905

119889119909119895

119889119905

) = 2119892119894119895

119889119909119894

119889119905

(119863

120575119909119895

119889119905

+ 119878119895

119901119896

119889119909119901

119889119905

120575119909119896

)

120575119904 = int

1199052

1199051

119892119894119895

119889119909119894

119889119904

119863120575119909119895

+ int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119904

119889119909119901

120575119909119896

= int

1199052

1199051

119863(119892119894119895

119889119909119894

119889119904

120575119909119895

)

minus int

1199052

1199051

119892119894119895119863

119889119909119894

119889119904

120575119909119895

+ int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119904

119889119909119901

120575119909119896

(46)


120575119904 = int

1199052

1199051

(119892119894119895119878119895

119901119896

119889119909119894

119889119905

119889119909119901

120575119909119896

minus 119892119894119895119863

119889119909119894

119889119905

120575119909119895

) (47)


int

1199052

1199051

(119892119894119895119878119895

119901119896

119889119909119894

119889119904

119889119909119901

120575119909119896

minus 119892119894119895119863

119889119909119894

119889119904

120575119909119895

) = 0 (48)


119892119894119896119878119896

119901119895

119889119909119894

119889119904

119889119909119901

minus 119892119894119895119863

119889119909119894

119889119904

= 0 (49)


119902

119892119895119902

119892119894119896119878119896

119901119895120585119894

119889119909119901


120575119904 = int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119905

119889119909119901

120575119909119896

(50)


119894119896and 119878

119895





119895



int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119905

119889119909119901

120575119909119896

(51)


119894119895


119896

= 119878119896

119894119895119860119894

119861119895


119894

120591 and 119861119895

120591


119901119902119878119901

119894119895119860119894

119861119895

119878119902

119896119897119860119896

119861119897

1205914


119894119896) with







119894119895


1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (52)

that is

Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (53)


Γ119896

119894119895=

1

2

119892119896119901

(119892119894119901119895119897

+ 119892119895119901119894

minus 119892119894119895119901

+ 119892119894119898119878119898

119895119901+ 119892119895119898119878119898

119894119901)

+

1

2

119878119896

119894119895

(54)



Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (55)



Γ119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

+ 119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

+

1

2

119878119901

119896119897

(56)


119875119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894) (57)


119901

119896119897+ 119872119901

119896119897 here 119872119896



119861119894

= 119886119860119894

(58)


119889119860119896

= minus (119875119896

119894119895+119872119896

119894119895)119860119894

119889119909119895

119889119861119896

= minus119875119896

119894119895119861119894

119889119909119895

(59)

Then we have

119860119896

119889119886 + 119886119889119860119896

= minus119875119896

119894119895119886119860119894

119889119909119895

(60)

119860119896

119889119886

119886

= 119872119896

119894119895119860119894

119889119909119895

(61)



119896119897+119872119901

119896119897


119889 ln 119886119889120591

119860119896

= 119872119896

119894119895119860119894

119860119895

(62)



119860119897

119872119896

119894119895119860119894

119860119895

minus 119860119896

119872119897

119894119895119860119894

119860119895

= 0 (63)

or

120575119897

119898119872119896

119894119895119860119894

119860119895

119860119898

minus 120575119896

119898119872119897

119894119895119860119894

119860119895

119860119898

= 0 (64)




120575119897

119898119872119896

119894119895minus 120575119896

119898119872119897

119894119895+ 120575119897

119894119872119896

119895119898minus 120575119896

119894119872119897

119895119898+ 120575119897

119895119872119896

119898119894minus 120575119896

119895119872119897

119898119894

= 0

(65)


have

119872119896

119894119895=

1

119899 + 1

(120575119896

119894119872119897

119895119897+ 120575119896

119895119872119897

119894119897) (66)



119894119895


119894119895 then let 119872119901

119896119897equiv

(12)119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) then119872119897

119896119897equiv (12)119878

119897

119896119897

We obtained

Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (67)



119894119895


119894119895=

(12)(1(119899 + 1))(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897



119896

119894119895119860119894

119889119909119895



119894119895







119894119895= 0 Assume that 119894 119872 sub 119875


119909rarr

119879119909119875|119872

rarr 119879perp



perp



119883119884 = 119879(nabla

1015840

119883119884) + perp (nabla

1015840

119883119884) Let it be

nabla119883119884 = 119879(nabla

1015840

119883119884) 120572(119883 119884) = perp (nabla




⟨1198771015840

(119883 119884)119885119882⟩ = ⟨119877 (119883 119884)119885119882⟩

+ ⟨120572 (119883 119885) 120572 (119884119882)⟩

minus ⟨120572 (119884 119885) 120572 (119883119882)⟩

(68)



(nabla10158401015840

119883120572) (119884 119885) = 119863

119883(120572 (119884 119885)) minus 120572 (nabla

119883119884119885)

minus 120572 (119884 nabla119883119885)

(69)

Codazzi equation is

perp ⟨1198771015840

(119883 119884)119885⟩ = (nabla10158401015840

119883120572) (119884 119885) minus (nabla

10158401015840

119884120572) (119883 119885) (70)





119909119894

= 119909119894

(1199101

119910119899minus1

) (71)


119886120572120573

= 119892119894119895

120597119909119894

120597119910120572

120597119909119895

120597119910120573

(72)


120572120573of



we assume that 119866120572120573120574



120572120573


119894119895and

119878119896

119894119895 we have

119866120572

120573120574

=

1

2

(119886120572120578

(119886120573120578120574

+ 119886120574120578120573

+ 119886120573120574120578

+ 119886120573120583119879120583

120574120578+ 119886120574120583119879120583

120573120578) + 119879120572

120574120573)

(73)


119866120572



1015840

by the formula

119866120572

120573120574= 1198661205721015840

12057310158401205741015840

120597119906120572

1205971199061205721015840

1205971199061205731015840

120597119906120573

1205971199061205741015840

120597119906120574+

120597119906120572

1205971199061205721015840

1205972

1199061205721015840

120597119906120573120597119906120574

(74)




119866120572

120573120574

120597119909119896

120597119906120572= Γ119896

119894119895

120597119909119894

120597119906120573

120597119909119895

120597119906120574+

1205972

119909119896

120597119906120573120597119906120574

(75)

We obtain

1

2

(119886120572120578

(119886120573120578120574

+ 119886120574120578120573

+ 119886120573120574120578

+ 119886120573120583119879120583

120574120578+ 119886120574120583119879120583

120573120578) + 119879120572

120574120573)

120597119909119896

120597119906120572

=

1

2

(119892119896119899

(119892119899119894119895

+ 119892119899119895119894

minus 119892119894119895119899

+ 119892119894119898119878119898

119895119899+ 119892119895119898119878119898

119894119899) + 119878119896

119894119895)

times

120597119909119894

120597119906120573

120597119909119895

120597119906120574+

1205972

119909119896

120597119906120573120597119906120574

(76)


119879120574

120572120573= 119886120574120578

119892119901119902119878119901

119894119895

120597119909119894

120597119910120572

120597119909119895

120597119910120573

120597119909119902

120597119910120578 (77)

using tensors 119886120572120573

and 119879120574












119860119894120572

119895120573120574= 119860119894120572

119895120573120574+ Γ119894

119901119896119860119901120572

119895120573

120597119909119896

120597119910120574minus Γ119901

119895119902119860119894120572

119901120573

120597119909119902

120597119910120574+ 119866120572

120578120574119860119894120578

119895120573

minus 119866120578

120573120574119860119894120572

119895120578

(78)


119906119894120572120573

minus 119906119894120573120572

= 119877119901

119896119897119894119906119901

120597119909119897

120597119910120572

120597119909119896

120597119910120573

+ 119878119902

119896119897119906119894119902

120597119909119897

120597119910120572

120597119909119896

120597119910120573

119906120574120572120573

minus 119906120574120573120572

= 119877120578

120573120572120574119906120578+ 119879120578

120573120572119906120574120578

(79)

where 119877120578



120573120574


120585119894

120572=

120597119909119894

120597119910120572 (80)



120585119894

1 120585

119894

119899minus1 ]119894 (81)

where 120585119894

1 120585

119894




120572

120585119894

120572120574= 120585119894

120572120574+ Γ119894

119901119902120585119901

120572

120597119909119902

120597119910120574minus 119866120578

120572120574120585119894

120578 (82)


120585119894

120572120574minus 120585119894

120574120572= 119878119894

119901119902120585119901

120572120585119902

120574+ 119879120578

120574120572120585119894

120578 (83)


0 = 119886120572120573120574

= (119892119894119895120585119894

120572120585119895

120573)

120574

= 119892119894119895120585119894

120572120574120585119895

120573+ 119892119894119895120585119894

120572120585119895

120573120574 (84)

we obtain

119892119894119895120585119894

120572120574120585119895

120573= 0 (85)


120585119894

120573120572= 120587120572120573]119894 (86)

Remark 11 120587120572120573



120574) the formula

119892119894119895]119894120574120585119894

120572= minus120587120574120572 (87)


]119894120574= minus119886120578120583

120587120583120574120585119894

120578 (88)



Further we obtain

120585119894

120573120594120582minus 120585119894

120573120582120594= minus119877119894

119896119897119901120585119896

120582120585119897

120594120585119901

120573+ 119877120590

120582120594120573120585119894

120590+ 119879120590

120582120594120585119894

120573120590

= (120587120594120573120582

minus 120587120582120573120594

) ]119894

minus (120587120594120573120587120578120582119886120578120590

minus 120587120582120573120587120578120594119886120578120590

) 120585119894

120590

(89)


120572 we have

119877120572120582120594120573

= 119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573120585119894

120572minus (120587120594120573120587120572120582

minus 120587120582120573120587120572120594) (90)


]119894120594120582

minus ]119894120582120594

= minus119877119894

119896119897119901120585119896

120582120585119897

120594]119901 + 119879

120590

120582120594]119894120590

= (120587120578120582120594

119886120578120590

minus 120587120578120594120582

119886120578120590

) 120585119894

120590

(91)


minus119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573]119894 + 119879

120590

120582120594120587120590120573

= 120587120594120573120582

minus 120587120582120573120594

(92)



minus119877119894119896119897119901

120585119896

120582120585119897

120594]119901120585119894120572+ 119879120590

120582120594120587120572120590

= 120587120572120582120594

minus 120587120572120594120582

(93)










119878119894

119895119896119901119902minus 119878119894

119895119896119902119901= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

+ 119878119905

119902119901119878119894

119895119896119905

(94)

or

119878119894

119895119896119901119902minus 119878119894

119895119896119902119901minus 119878119905

119902119901119878119894

119895119896119905

= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

(95)


119878119894

119895119896119901119894minus 119878119894

119895119896119894119901minus 119878119905

119894119901119878119894

119895119896119905

= (119878119894

119895119896119901minus 119878119894

119902119901119878119902

119895119896)

119894

minus 119878119894

119895119896119894119901minus 119878119894

119901119902119894119878119902

119895119896

(96)


(119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896)119894

minus 119892119896119901

119892119895119904

119878119894

119904119896119894119901minus 119892119896119901

119892119895119904

119878119894

119901119902119894119878119902

119904119896

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905

minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(97)


119867119895119894

= 119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896

119865119895119901

= 119892119896119901

119892119895119904

119878119894

119904119896119894

(98)


119867119895119894

119894minus 119865119895119894

119894minus 119892119896119901

119892119895119904

119878119902

119904119896119865119901119902

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(99)

where 119865119901119902

= 119878119894

119901119902119894


119894119896119897+ 119877119901

119896119897119894+ 119877119901

119897119894119896= 0 hence

119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896+ 119878119901

119897119902119878119902

119894119896+ 119878119901

119896119902119878119902

119897119894+ 119878119901

119894119902119878119902

119896119897= 0 (100)


119878119901

119894119896119901+ 119878119901

119896119901119894+ 119878119901

119901119894119896= 0 (101)




119894119895119878119902


expression

119878119901

119894119895119901= 120593119894119895minus 120593119895119894 (102)

Next if we put 119878119901119894119895119901


=



119894 System (99)

takes the form

119867119895119894

119894= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

119865119894119895

= 0

(103)


119862119894119895119896

= 119892119901119895

119892119902119896

119878119894

119901119902+ 119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896

119901119902 (104)


119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+ 119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904

minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

(105)



119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

=

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

(106)

hence

119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902) (107)


119862119894119896119895

119894= minus119862119894119895119896

119894= minus (119862

119894119895119896

119894+ Γ119895

119901119894119862119894119901119896

+ Γ119896

119901119894119862119894119895119901

+ Γ119894

119901119894119862119901119896119895

) (108)


= 119892119901119895

119892119902119896

119878119894

119901119902+

119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896


Γ119895

119901119894119862119894119901119896

= Γ119895

119894119901119862119894119901119896

=

1

2

(Γ119895

119894119901119862119894119901119896

+ Γ119895

119901119894119862119901119894119896

) =

1

2

119878119895

119894119901119862119901119896119894

= minus

1

2

119878119895

119901119894119862119896119901119894

(109)

similarly we obtain

Γ119896

119901119894119862119894119895119901

= Γ119896

119894119901119862119901119895119894

=

1

2

119862119895119901119894

(Γ119896

119894119901minus Γ119896

119901119894) =

1

2

119878119896

119894119901119862119895119901119894

=

1

2

119878119896

119901119902119862119895119902119901

(110)

Then we write

119862119894119896119895

119894= minus119862119894119895119896

119894

= minus119862119894119895119896

119894minus

1

2

119878119895

119901119902119862119896119901119902

+

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894+

1

2

119878119895

119901119902119862119896119901119902

minus

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

(111)



119901

119901119897=

(12)119892119894119901119897119892119894119901

= (1radic119892)(120597radic119892120597119909119897

) and Γ119901

119897119901= Γ119901

119901119897+ 119878119901

119897119901and

obtain

Γ119901

119901119897=

1

radicminus119892


120597119909119897

+ (ln120595)119897= (ln (120595radicminus119892))

119897 (112)

Then we obtain

119867119895119896

minus 119867119896119895

minus 119865119895119896

= minus119862119894119895119896


119894119862119894119896119895

(113)


120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


119894+ (ln (120595radicminus119892))

119894119862119894119896119895

)

120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


)119894

(114)


(120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

))119896

= 0 (115)







119894

119895119896 we



119894

119895119896and 119892

119894119896


119877119894119895= Γ119901

119894119901119895minus Γ119901

119894119895119901+ Γ119901

119902119895Γ119902

119894119901minus Γ119901

119902119901Γ119902

119894119895 (116)


119894119897119878119897

119896119895)119892119894119896


Γ119894

119895119896minus Γ119894


int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881 (117)




119877119894119896+ 119878119895

119894119897119878119897

119896119895= 0 (118)


119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901

minus 120575119895

119896(119892119894119901

119901minus

1

2

119892119894119901

119892119898119899119892119898119899

119901+ 119892119901119902

Γ119894

119901119902) minus 119892119894119895

Γ119901

119896119901

+ 2 (119892119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(119)



minus3119892119894119895

119895+

3

2

119892119894119895

119892119901119902119892119901119902

119895minus 3119892119901119902

Γ119894

119901119902+ 119892119894119901

119878119895

119901119895= 0 (120)

Therefore we have

119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901minus 119892119894119895

Γ119901

119896119901

minus

1

3

120575119895

119896119892119894119901

119878119902

119901119902+ 2 (119892

119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(121)


minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901+ 2 (119892

119901119895119878119901

119894119896+ 119892119894119901119878119901

119896119895) = 0

(122)



5 (119892119894119901119878119901

119896119895+ 119892119895119901119878119901

119894119896) +

1

3

(119892119895119896119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0 (123)

Further we obtained

minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901minus

2

15

(119892119896119895119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0

(124)


0 then from (123) we have 119878119894




119898+

119882119892) = 0 where 119882

119898and 119882




120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int (119877119894119896radicminus119892120575119892

119894119896

+ 119877119894119896119892119894119896

120575radicminus119892 + 119892119894119896

radicminus119892120575119877119894119896

+ 119878119895

119894119897119878119897

119896119895radicminus119892120575119892

119894119896

+ 119878119895

119894119897119878119897

119896119895119892119894119896

120575radicminus119892) 119889119881

119877119894119896119892119894119896


1

2

119877119901119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(125)

Similarly we obtain

119878119895

119894119897119878119897

119896119895119892119894119896


1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(126)



119894119896

(120575Γ119897

119896119894)119897minus 119892119894119896

(120575Γ119897

119896119897)119894= (119892119894119896

120575Γ119897

119896119894minus 119892119894119897

120575Γ119901

119897119901)119897



119894119896= 119892119894119896

120575(Γ119901

119902119901Γ119902

119896119894minus

Γ119901

119902119894Γ119902119897



119892119894119896

120575119877119894119896= (119878119901

119894119901119878119902

119902119896minus 119878119901

119894119902119878119902

119901119896minus 119892119896119901119892119902119905

119878119898

119894119905119878119901

119902119898) 120575119892119894119896

(127)

Thus we have

120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(128)


120575119882119892

= 1198701int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= 1198701int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(129)


1198883


120575119882119892= 1198702int119879119894119896radicminus119892120575119892

119894119896

119889119881 (130)


2usually


119892+120575119882119898=

0 we find relations

int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902


119894119896

119889119881 = 0

(131)



119877119894119896minus

1

2

119892119894119896119877 + 119878119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902+ 119878119895

119894119897119878119897

119896119895

minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896= 119870119879119894119896

(132)


2

7 Conclusions







References




























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





We rewrite 119889(119892119894119896119860119894

119861119896

) = 0 as (119892119894119896119897

minus 119892119898119896Γ119898

119894119897minus

119892119894119898Γ119898

119896119897)119860119894

119861119896

119889119909119897


119901119897119860119901

119889119909119897




equiv

(120597120597119909119897



119892119894119896119897

= 119892119898119896Γ119898

119894119897+ 119892119894119898Γ119898

119896119897

119892119897119894119896

= 119892119898119894Γ119898

119897119896+ 119892119897119898Γ119898

119894119896

119892119896119897119894

= 119892119898119897Γ119898

119896119894+ 119892119896119898Γ119898

119897119894

(10)


119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

= 119892119898119896119878119898

119894119897+ 119892119898119897119878119898

119894119896+ 119892119894119898Γ119898

119896119897+ 119892119898119894Γ119898

119897119896 (11)

where 119878119898119894119897= Γ119898

119894119897minus Γ119898


119892119894119898(Γ119898

119896119897+ Γ119898

119897119896)

= 119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

+ 119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894

Γ119901

119896119897+ Γ119901

119897119896

= 119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

+ 119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

(12)


119896119897minus

Γ119901

119897119896= 119878119901


Γ119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

+ 119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

+

1

2

119878119901

119896119897

(13)


119875119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894) (14)


119871119901

119896119897equiv

1

2

119878119901

119896119897+

1

2

119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) (15)


119901



119898

119894119896


119875119901


119894119896


119894119896and the tensor 119878119898

119896119897 namely

Γ119901

119896119897= 119875119901

119896119897+ 119871119901

119896119897 (16)



symmetric (12)119892119901119894(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) and torsion (12)119878119901

119896119897


Γ119901

119901119897=

1

2

119892119894119901119897119892119894119901

=

1

radic119892

120597radic119892

120597119909119897

where 119892 = det 1003816100381610038161003816119892119894119896

1003816100381610038161003816 (17)


119889119860119894

= minusΓ119894

119901119897119860119901

119889119909119897

(18)




119906119894119897equiv 119906119894119897minus Γ119896

119894119897119906119896

119906119894

119897equiv 119906119894

119897+ Γ119894

119896119897119906119896

(19)



119894119897119906119896 (20)



119885119896

= 119878119896

119894119895119860119894

119861119895

1205912

(21)


119895



119906119894119897119896

minus 119906119894119896119897

= 119877119901

119896119897119894119906119901+ 119878119902

119896119897119906119894119902 (22)

where we identified

119877119901

119896119897119894equiv Γ119901

119894119896119897minus Γ119901

119894119897119896+ Γ119901

119902119897Γ119902

119894119896minus Γ119901

119902119896Γ119902

119894119897 (23)

119877119901


119906119894

119897119896minus 119906119894

119896119897= minus119877119894

119896119897119901119906119901

+ 119878119902

119896119897119906119894

119902 (24)


119863119860119894

equivsim 119889119860119894

+ Γ119894

119895119896119860119895

119889119909119896

= (119860119894

119896+ Γ119894

119895119896119860119895

) 119889119909119896

(25)



119863119863119860119894

minus 119863119863119860119894

= minus119877119894

119896119897119901119860119901 119889119909119896

119889119909119897

(26)


119889119909119897



119877119901

119894119896119897= Γ119901

119897119894119896minus Γ119901

119897119896119894+ Γ119901

119902119896Γ119902

119897119894minus Γ119901

119902119894Γ119902

119897119896 (27)


119877119901

119894119896119897= 119875119901

119897119894119896+ 119871119901

119897119894119896minus 119875119901

119897119896119894+ 119871119901

119897119896119894+ (119875119901

119902119896+ 119871119901

119902119896) (119875119901

119897119894+ 119871119902

119897119894)

minus (119875119901

119902119894+ 119871119901

119902119894) (119875119901

119897119896+ 119871119902

119897119896)

= 119875119901

119897119894119896minus 119875119901

119897119896119894+ 119875119901

119902119896119875119902

119897119894minus 119875119901

119902119894119875119902

119897119896+ 119871119901

119897119894119896minus 119871119901

119897119896119894

+ 119875119901

119902119896119871119902

119897119894+ 119875119902

119897119894119871119901

119902119896minus 119875119901

119902119894119871119902

119897119896minus 119875119902

119897119896119871119901

119902119894

+ 119871119901

119902119896119871119902

119897119894minus 119871119901

119902119894119871119902

119897119896

(28)


119875119901

119894119896119897equiv 119875119901

119897119894119896minus 119875119901

119897119896119894+ 119875119901

119902119896119875119902

119897119894+ 119875119901

119902119894119875119902

119897119896(29)


119885119901

119894119896119897equiv 119871119901

119902119896119871119902

119897119894minus 119871119901

119902119894119871119902

119897119896(30)

is tensor and

119879119901

119894119896119897equiv 119871119901

119897119894119896minus 119871119901

119897119896119894+ 119875119901

119902119896119871119902

119897119894+ 119875119902

119897119894119871119901

119902119896minus 119875119901

119902119894119871119902

119897119896minus 119875119902

119897119896119871119901

119902119894(31)




119879119901

119894119896119897= 119871119901

119897119894119896minus 119871119901

119897119896119894minus 119871119902

119897119894Γ119901

119902119896+ 119871119901

119902119894Γ119902

119897119896+ 119871119901

119897119902Γ119902

119894119896+ 119871119902

119897119896Γ119901

119902119894

minus 119871119901

119902119896Γ119902

119897119894minus 119871119901

119897119902Γ119902

119896119894+ 119875119901

119902119896119871119902

119897119894+ 119875119902

119897119894119871119902

119902119896

minus 119875119901

119902119894119871119902

119897119896minus 119875119902

119897119896119871119901

119902119894

= 119871119901

119897119894119896minus 119871119901

119897119896119894minus 119871119902

119897119894119871119901

119902119896+ 119871119901

119902119894119871119902

119897119896+ 119871119902

119897119896119871119901

119902119894

minus 119871119901

119902119896119871119902

119897119894+ 119871119901

119897119902119878119902

119894119896

(32)



119872119901

119894119896119897equiv 119879119901

119894119896119897+ 119885119901

119894119896119897 (33)

then we obtain

119872119901

119894119896119897= 119871119901

119897119894119896minus 119871119901

119897119896119894+ 119871119901

119897119902119878119902

119894119896+ 119871119901

119902119894119871119902

119897119896minus 119871119901

119902119896119871119902

119897119894 (34)


119877119901

119894119896119897= 119875119901

119894119896119897+119872119901

119894119896119897 (35)




119895119896


119877119901

119894119896119897+ 119877119901

119896119897119894+ 119877119901

119897119894119896= 119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896+ Γ119901

119902119896119878119902

119897119894+ Γ119901

119902119896119878119902

119894119896+ Γ119901

119902119894119878119902

119896119897

= 119878119901

119894119896119897+ Γ119902

119894119897119878119901

119902119896+ Γ119902

119896119897119878119901

119894119902+ 119878119901

119896119897119894+ Γ119902

119896119894119878119901

119902119897+ Γ119902

119897119894119878119901

119896119902

+ 119878119901

119897119894119896+ Γ119902

119897119896119878119901

119902119894+ Γ119902

119894119896119878119901

119897119902

= 119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896

+ 119878119901

119897119902119878119902

119894119896+ 119878119901

119896119902119878119902

119897119894+ 119878119901

119894119902119878119902

119896119897

(36)


119878119894

119895119901119878119901

119896119894+ 119878119894

119896119901119878119901

119894119895= 0

119878119894

119894119901119878119901

119895119896= 0

(37)


119878119894

119895119901119878119901

119896119894+ 119878119894

119896119901119878119901

119894119895+ 119878119894

119894119901119878119901

119895119896= 0 (38)




1198751199011015840

11989610158401198971015840= 119875119901

119896119897

1205971199091199011015840

120597119909119901

120597119909119896

1205971199091198961015840

120597119909119897

1205971199091198971015840+

1205972

119909119901

1205971199091198961015840

1205971199091198971015840

1205971199091199011015840

120597119909119901 (39)


119895119896equiv Γ119894

119895119896minus Γ119894

119896119895)




Value

119872119901

119896119897equiv

1

2

119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) (40)




1198892

119909119896

1198891205912= minusΓ119896

119894119895

119889119909119894

119889120591

119889119909119895

119889120591

(41)


119896119897+119872119901

119896119897 In (6) 120591


1198892

119909119896

1198891199042

= minusΓ119896

119894119895

119889119909119894

119889119904

119889119909119895

119889119904

(42)




119875119901

119896119897+119872119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894)

+

1

2

119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

(43)



1198892

119909119896

1198891199042

= minusΓ119896

119894119895

119889119909119894

119889119904

119889119909119895

119889119904

(44)



= 119909119894

(119905) 119905 isin [1199051 1199052] We


120575119878 = int

1199052

1199051

120575radic119892119894119895

119889119909119894

119889119905

119889119909119895

119889119905

119889119905

= int

1199052

1199051

120575 (119892119894119895(119889119909119894

119889119905) (119889119909119895

119889119905))

2radic119892119894119895(119889119909119894119889119905) (119889119909

119895119889119905)

119889119905

120575 (119892119894119895

119889119909119894

119889119905

119889119909119895

119889119905

) = 119892119894119895119863

119889119909119894

119889119905

119889119909119895

119889119905

+ 119892119894119895

119889119909119894

119889119905

119863

119889119909119895

119889119905

= 2119892119894119895

119889119909119894

119889119905

119863

119889119909119895

119889119905

119863

119889119909119895

119889119905

= 120575

119889119909119895

119889119905

+ Γ119895

119901119896

119889119909119901

119889119905

120575119909119896

119863

120575119909119895

119889119905

=

119889

119889119905

120575119909119895

+ Γ119895

119896119901

119889119909119896

119889119905

120575119909119901

119863

119889119909119895

119889119905

= 119863

120575119909119895

119889119905

+ 119878119895

119901119896

119889119909119901

119889119905

120575119909119896

(45)


120575(119892119894119895

119889119909119894

119889119905

119889119909119895

119889119905

) = 2119892119894119895

119889119909119894

119889119905

(119863

120575119909119895

119889119905

+ 119878119895

119901119896

119889119909119901

119889119905

120575119909119896

)

120575119904 = int

1199052

1199051

119892119894119895

119889119909119894

119889119904

119863120575119909119895

+ int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119904

119889119909119901

120575119909119896

= int

1199052

1199051

119863(119892119894119895

119889119909119894

119889119904

120575119909119895

)

minus int

1199052

1199051

119892119894119895119863

119889119909119894

119889119904

120575119909119895

+ int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119904

119889119909119901

120575119909119896

(46)


120575119904 = int

1199052

1199051

(119892119894119895119878119895

119901119896

119889119909119894

119889119905

119889119909119901

120575119909119896

minus 119892119894119895119863

119889119909119894

119889119905

120575119909119895

) (47)


int

1199052

1199051

(119892119894119895119878119895

119901119896

119889119909119894

119889119904

119889119909119901

120575119909119896

minus 119892119894119895119863

119889119909119894

119889119904

120575119909119895

) = 0 (48)


119892119894119896119878119896

119901119895

119889119909119894

119889119904

119889119909119901

minus 119892119894119895119863

119889119909119894

119889119904

= 0 (49)


119902

119892119895119902

119892119894119896119878119896

119901119895120585119894

119889119909119901


120575119904 = int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119905

119889119909119901

120575119909119896

(50)


119894119896and 119878

119895





119895



int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119905

119889119909119901

120575119909119896

(51)


119894119895


119896

= 119878119896

119894119895119860119894

119861119895


119894

120591 and 119861119895

120591


119901119902119878119901

119894119895119860119894

119861119895

119878119902

119896119897119860119896

119861119897

1205914


119894119896) with







119894119895


1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (52)

that is

Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (53)


Γ119896

119894119895=

1

2

119892119896119901

(119892119894119901119895119897

+ 119892119895119901119894

minus 119892119894119895119901

+ 119892119894119898119878119898

119895119901+ 119892119895119898119878119898

119894119901)

+

1

2

119878119896

119894119895

(54)



Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (55)



Γ119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

+ 119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

+

1

2

119878119901

119896119897

(56)


119875119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894) (57)


119901

119896119897+ 119872119901

119896119897 here 119872119896



119861119894

= 119886119860119894

(58)


119889119860119896

= minus (119875119896

119894119895+119872119896

119894119895)119860119894

119889119909119895

119889119861119896

= minus119875119896

119894119895119861119894

119889119909119895

(59)

Then we have

119860119896

119889119886 + 119886119889119860119896

= minus119875119896

119894119895119886119860119894

119889119909119895

(60)

119860119896

119889119886

119886

= 119872119896

119894119895119860119894

119889119909119895

(61)



119896119897+119872119901

119896119897


119889 ln 119886119889120591

119860119896

= 119872119896

119894119895119860119894

119860119895

(62)



119860119897

119872119896

119894119895119860119894

119860119895

minus 119860119896

119872119897

119894119895119860119894

119860119895

= 0 (63)

or

120575119897

119898119872119896

119894119895119860119894

119860119895

119860119898

minus 120575119896

119898119872119897

119894119895119860119894

119860119895

119860119898

= 0 (64)




120575119897

119898119872119896

119894119895minus 120575119896

119898119872119897

119894119895+ 120575119897

119894119872119896

119895119898minus 120575119896

119894119872119897

119895119898+ 120575119897

119895119872119896

119898119894minus 120575119896

119895119872119897

119898119894

= 0

(65)


have

119872119896

119894119895=

1

119899 + 1

(120575119896

119894119872119897

119895119897+ 120575119896

119895119872119897

119894119897) (66)



119894119895


119894119895 then let 119872119901

119896119897equiv

(12)119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) then119872119897

119896119897equiv (12)119878

119897

119896119897

We obtained

Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (67)



119894119895


119894119895=

(12)(1(119899 + 1))(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897



119896

119894119895119860119894

119889119909119895



119894119895







119894119895= 0 Assume that 119894 119872 sub 119875


119909rarr

119879119909119875|119872

rarr 119879perp



perp



119883119884 = 119879(nabla

1015840

119883119884) + perp (nabla

1015840

119883119884) Let it be

nabla119883119884 = 119879(nabla

1015840

119883119884) 120572(119883 119884) = perp (nabla




⟨1198771015840

(119883 119884)119885119882⟩ = ⟨119877 (119883 119884)119885119882⟩

+ ⟨120572 (119883 119885) 120572 (119884119882)⟩

minus ⟨120572 (119884 119885) 120572 (119883119882)⟩

(68)



(nabla10158401015840

119883120572) (119884 119885) = 119863

119883(120572 (119884 119885)) minus 120572 (nabla

119883119884119885)

minus 120572 (119884 nabla119883119885)

(69)

Codazzi equation is

perp ⟨1198771015840

(119883 119884)119885⟩ = (nabla10158401015840

119883120572) (119884 119885) minus (nabla

10158401015840

119884120572) (119883 119885) (70)





119909119894

= 119909119894

(1199101

119910119899minus1

) (71)


119886120572120573

= 119892119894119895

120597119909119894

120597119910120572

120597119909119895

120597119910120573

(72)


120572120573of



we assume that 119866120572120573120574



120572120573


119894119895and

119878119896

119894119895 we have

119866120572

120573120574

=

1

2

(119886120572120578

(119886120573120578120574

+ 119886120574120578120573

+ 119886120573120574120578

+ 119886120573120583119879120583

120574120578+ 119886120574120583119879120583

120573120578) + 119879120572

120574120573)

(73)


119866120572



1015840

by the formula

119866120572

120573120574= 1198661205721015840

12057310158401205741015840

120597119906120572

1205971199061205721015840

1205971199061205731015840

120597119906120573

1205971199061205741015840

120597119906120574+

120597119906120572

1205971199061205721015840

1205972

1199061205721015840

120597119906120573120597119906120574

(74)




119866120572

120573120574

120597119909119896

120597119906120572= Γ119896

119894119895

120597119909119894

120597119906120573

120597119909119895

120597119906120574+

1205972

119909119896

120597119906120573120597119906120574

(75)

We obtain

1

2

(119886120572120578

(119886120573120578120574

+ 119886120574120578120573

+ 119886120573120574120578

+ 119886120573120583119879120583

120574120578+ 119886120574120583119879120583

120573120578) + 119879120572

120574120573)

120597119909119896

120597119906120572

=

1

2

(119892119896119899

(119892119899119894119895

+ 119892119899119895119894

minus 119892119894119895119899

+ 119892119894119898119878119898

119895119899+ 119892119895119898119878119898

119894119899) + 119878119896

119894119895)

times

120597119909119894

120597119906120573

120597119909119895

120597119906120574+

1205972

119909119896

120597119906120573120597119906120574

(76)


119879120574

120572120573= 119886120574120578

119892119901119902119878119901

119894119895

120597119909119894

120597119910120572

120597119909119895

120597119910120573

120597119909119902

120597119910120578 (77)

using tensors 119886120572120573

and 119879120574












119860119894120572

119895120573120574= 119860119894120572

119895120573120574+ Γ119894

119901119896119860119901120572

119895120573

120597119909119896

120597119910120574minus Γ119901

119895119902119860119894120572

119901120573

120597119909119902

120597119910120574+ 119866120572

120578120574119860119894120578

119895120573

minus 119866120578

120573120574119860119894120572

119895120578

(78)


119906119894120572120573

minus 119906119894120573120572

= 119877119901

119896119897119894119906119901

120597119909119897

120597119910120572

120597119909119896

120597119910120573

+ 119878119902

119896119897119906119894119902

120597119909119897

120597119910120572

120597119909119896

120597119910120573

119906120574120572120573

minus 119906120574120573120572

= 119877120578

120573120572120574119906120578+ 119879120578

120573120572119906120574120578

(79)

where 119877120578



120573120574


120585119894

120572=

120597119909119894

120597119910120572 (80)



120585119894

1 120585

119894

119899minus1 ]119894 (81)

where 120585119894

1 120585

119894




120572

120585119894

120572120574= 120585119894

120572120574+ Γ119894

119901119902120585119901

120572

120597119909119902

120597119910120574minus 119866120578

120572120574120585119894

120578 (82)


120585119894

120572120574minus 120585119894

120574120572= 119878119894

119901119902120585119901

120572120585119902

120574+ 119879120578

120574120572120585119894

120578 (83)


0 = 119886120572120573120574

= (119892119894119895120585119894

120572120585119895

120573)

120574

= 119892119894119895120585119894

120572120574120585119895

120573+ 119892119894119895120585119894

120572120585119895

120573120574 (84)

we obtain

119892119894119895120585119894

120572120574120585119895

120573= 0 (85)


120585119894

120573120572= 120587120572120573]119894 (86)

Remark 11 120587120572120573



120574) the formula

119892119894119895]119894120574120585119894

120572= minus120587120574120572 (87)


]119894120574= minus119886120578120583

120587120583120574120585119894

120578 (88)



Further we obtain

120585119894

120573120594120582minus 120585119894

120573120582120594= minus119877119894

119896119897119901120585119896

120582120585119897

120594120585119901

120573+ 119877120590

120582120594120573120585119894

120590+ 119879120590

120582120594120585119894

120573120590

= (120587120594120573120582

minus 120587120582120573120594

) ]119894

minus (120587120594120573120587120578120582119886120578120590

minus 120587120582120573120587120578120594119886120578120590

) 120585119894

120590

(89)


120572 we have

119877120572120582120594120573

= 119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573120585119894

120572minus (120587120594120573120587120572120582

minus 120587120582120573120587120572120594) (90)


]119894120594120582

minus ]119894120582120594

= minus119877119894

119896119897119901120585119896

120582120585119897

120594]119901 + 119879

120590

120582120594]119894120590

= (120587120578120582120594

119886120578120590

minus 120587120578120594120582

119886120578120590

) 120585119894

120590

(91)


minus119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573]119894 + 119879

120590

120582120594120587120590120573

= 120587120594120573120582

minus 120587120582120573120594

(92)



minus119877119894119896119897119901

120585119896

120582120585119897

120594]119901120585119894120572+ 119879120590

120582120594120587120572120590

= 120587120572120582120594

minus 120587120572120594120582

(93)










119878119894

119895119896119901119902minus 119878119894

119895119896119902119901= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

+ 119878119905

119902119901119878119894

119895119896119905

(94)

or

119878119894

119895119896119901119902minus 119878119894

119895119896119902119901minus 119878119905

119902119901119878119894

119895119896119905

= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

(95)


119878119894

119895119896119901119894minus 119878119894

119895119896119894119901minus 119878119905

119894119901119878119894

119895119896119905

= (119878119894

119895119896119901minus 119878119894

119902119901119878119902

119895119896)

119894

minus 119878119894

119895119896119894119901minus 119878119894

119901119902119894119878119902

119895119896

(96)


(119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896)119894

minus 119892119896119901

119892119895119904

119878119894

119904119896119894119901minus 119892119896119901

119892119895119904

119878119894

119901119902119894119878119902

119904119896

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905

minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(97)


119867119895119894

= 119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896

119865119895119901

= 119892119896119901

119892119895119904

119878119894

119904119896119894

(98)


119867119895119894

119894minus 119865119895119894

119894minus 119892119896119901

119892119895119904

119878119902

119904119896119865119901119902

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(99)

where 119865119901119902

= 119878119894

119901119902119894


119894119896119897+ 119877119901

119896119897119894+ 119877119901

119897119894119896= 0 hence

119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896+ 119878119901

119897119902119878119902

119894119896+ 119878119901

119896119902119878119902

119897119894+ 119878119901

119894119902119878119902

119896119897= 0 (100)


119878119901

119894119896119901+ 119878119901

119896119901119894+ 119878119901

119901119894119896= 0 (101)




119894119895119878119902


expression

119878119901

119894119895119901= 120593119894119895minus 120593119895119894 (102)

Next if we put 119878119901119894119895119901


=



119894 System (99)

takes the form

119867119895119894

119894= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

119865119894119895

= 0

(103)


119862119894119895119896

= 119892119901119895

119892119902119896

119878119894

119901119902+ 119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896

119901119902 (104)


119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+ 119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904

minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

(105)



119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

=

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

(106)

hence

119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902) (107)


119862119894119896119895

119894= minus119862119894119895119896

119894= minus (119862

119894119895119896

119894+ Γ119895

119901119894119862119894119901119896

+ Γ119896

119901119894119862119894119895119901

+ Γ119894

119901119894119862119901119896119895

) (108)


= 119892119901119895

119892119902119896

119878119894

119901119902+

119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896


Γ119895

119901119894119862119894119901119896

= Γ119895

119894119901119862119894119901119896

=

1

2

(Γ119895

119894119901119862119894119901119896

+ Γ119895

119901119894119862119901119894119896

) =

1

2

119878119895

119894119901119862119901119896119894

= minus

1

2

119878119895

119901119894119862119896119901119894

(109)

similarly we obtain

Γ119896

119901119894119862119894119895119901

= Γ119896

119894119901119862119901119895119894

=

1

2

119862119895119901119894

(Γ119896

119894119901minus Γ119896

119901119894) =

1

2

119878119896

119894119901119862119895119901119894

=

1

2

119878119896

119901119902119862119895119902119901

(110)

Then we write

119862119894119896119895

119894= minus119862119894119895119896

119894

= minus119862119894119895119896

119894minus

1

2

119878119895

119901119902119862119896119901119902

+

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894+

1

2

119878119895

119901119902119862119896119901119902

minus

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

(111)



119901

119901119897=

(12)119892119894119901119897119892119894119901

= (1radic119892)(120597radic119892120597119909119897

) and Γ119901

119897119901= Γ119901

119901119897+ 119878119901

119897119901and

obtain

Γ119901

119901119897=

1

radicminus119892


120597119909119897

+ (ln120595)119897= (ln (120595radicminus119892))

119897 (112)

Then we obtain

119867119895119896

minus 119867119896119895

minus 119865119895119896

= minus119862119894119895119896


119894119862119894119896119895

(113)


120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


119894+ (ln (120595radicminus119892))

119894119862119894119896119895

)

120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


)119894

(114)


(120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

))119896

= 0 (115)







119894

119895119896 we



119894

119895119896and 119892

119894119896


119877119894119895= Γ119901

119894119901119895minus Γ119901

119894119895119901+ Γ119901

119902119895Γ119902

119894119901minus Γ119901

119902119901Γ119902

119894119895 (116)


119894119897119878119897

119896119895)119892119894119896


Γ119894

119895119896minus Γ119894


int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881 (117)




119877119894119896+ 119878119895

119894119897119878119897

119896119895= 0 (118)


119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901

minus 120575119895

119896(119892119894119901

119901minus

1

2

119892119894119901

119892119898119899119892119898119899

119901+ 119892119901119902

Γ119894

119901119902) minus 119892119894119895

Γ119901

119896119901

+ 2 (119892119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(119)



minus3119892119894119895

119895+

3

2

119892119894119895

119892119901119902119892119901119902

119895minus 3119892119901119902

Γ119894

119901119902+ 119892119894119901

119878119895

119901119895= 0 (120)

Therefore we have

119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901minus 119892119894119895

Γ119901

119896119901

minus

1

3

120575119895

119896119892119894119901

119878119902

119901119902+ 2 (119892

119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(121)


minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901+ 2 (119892

119901119895119878119901

119894119896+ 119892119894119901119878119901

119896119895) = 0

(122)



5 (119892119894119901119878119901

119896119895+ 119892119895119901119878119901

119894119896) +

1

3

(119892119895119896119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0 (123)

Further we obtained

minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901minus

2

15

(119892119896119895119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0

(124)


0 then from (123) we have 119878119894




119898+

119882119892) = 0 where 119882

119898and 119882




120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int (119877119894119896radicminus119892120575119892

119894119896

+ 119877119894119896119892119894119896

120575radicminus119892 + 119892119894119896

radicminus119892120575119877119894119896

+ 119878119895

119894119897119878119897

119896119895radicminus119892120575119892

119894119896

+ 119878119895

119894119897119878119897

119896119895119892119894119896

120575radicminus119892) 119889119881

119877119894119896119892119894119896


1

2

119877119901119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(125)

Similarly we obtain

119878119895

119894119897119878119897

119896119895119892119894119896


1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(126)



119894119896

(120575Γ119897

119896119894)119897minus 119892119894119896

(120575Γ119897

119896119897)119894= (119892119894119896

120575Γ119897

119896119894minus 119892119894119897

120575Γ119901

119897119901)119897



119894119896= 119892119894119896

120575(Γ119901

119902119901Γ119902

119896119894minus

Γ119901

119902119894Γ119902119897



119892119894119896

120575119877119894119896= (119878119901

119894119901119878119902

119902119896minus 119878119901

119894119902119878119902

119901119896minus 119892119896119901119892119902119905

119878119898

119894119905119878119901

119902119898) 120575119892119894119896

(127)

Thus we have

120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(128)


120575119882119892

= 1198701int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= 1198701int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(129)


1198883


120575119882119892= 1198702int119879119894119896radicminus119892120575119892

119894119896

119889119881 (130)


2usually


119892+120575119882119898=

0 we find relations

int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902


119894119896

119889119881 = 0

(131)



119877119894119896minus

1

2

119892119894119896119877 + 119878119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902+ 119878119895

119894119897119878119897

119896119895

minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896= 119870119879119894119896

(132)


2

7 Conclusions







References




























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119863119863119860119894

minus 119863119863119860119894

= minus119877119894

119896119897119901119860119901 119889119909119896

119889119909119897

(26)


119889119909119897



119877119901

119894119896119897= Γ119901

119897119894119896minus Γ119901

119897119896119894+ Γ119901

119902119896Γ119902

119897119894minus Γ119901

119902119894Γ119902

119897119896 (27)


119877119901

119894119896119897= 119875119901

119897119894119896+ 119871119901

119897119894119896minus 119875119901

119897119896119894+ 119871119901

119897119896119894+ (119875119901

119902119896+ 119871119901

119902119896) (119875119901

119897119894+ 119871119902

119897119894)

minus (119875119901

119902119894+ 119871119901

119902119894) (119875119901

119897119896+ 119871119902

119897119896)

= 119875119901

119897119894119896minus 119875119901

119897119896119894+ 119875119901

119902119896119875119902

119897119894minus 119875119901

119902119894119875119902

119897119896+ 119871119901

119897119894119896minus 119871119901

119897119896119894

+ 119875119901

119902119896119871119902

119897119894+ 119875119902

119897119894119871119901

119902119896minus 119875119901

119902119894119871119902

119897119896minus 119875119902

119897119896119871119901

119902119894

+ 119871119901

119902119896119871119902

119897119894minus 119871119901

119902119894119871119902

119897119896

(28)


119875119901

119894119896119897equiv 119875119901

119897119894119896minus 119875119901

119897119896119894+ 119875119901

119902119896119875119902

119897119894+ 119875119901

119902119894119875119902

119897119896(29)


119885119901

119894119896119897equiv 119871119901

119902119896119871119902

119897119894minus 119871119901

119902119894119871119902

119897119896(30)

is tensor and

119879119901

119894119896119897equiv 119871119901

119897119894119896minus 119871119901

119897119896119894+ 119875119901

119902119896119871119902

119897119894+ 119875119902

119897119894119871119901

119902119896minus 119875119901

119902119894119871119902

119897119896minus 119875119902

119897119896119871119901

119902119894(31)




119879119901

119894119896119897= 119871119901

119897119894119896minus 119871119901

119897119896119894minus 119871119902

119897119894Γ119901

119902119896+ 119871119901

119902119894Γ119902

119897119896+ 119871119901

119897119902Γ119902

119894119896+ 119871119902

119897119896Γ119901

119902119894

minus 119871119901

119902119896Γ119902

119897119894minus 119871119901

119897119902Γ119902

119896119894+ 119875119901

119902119896119871119902

119897119894+ 119875119902

119897119894119871119902

119902119896

minus 119875119901

119902119894119871119902

119897119896minus 119875119902

119897119896119871119901

119902119894

= 119871119901

119897119894119896minus 119871119901

119897119896119894minus 119871119902

119897119894119871119901

119902119896+ 119871119901

119902119894119871119902

119897119896+ 119871119902

119897119896119871119901

119902119894

minus 119871119901

119902119896119871119902

119897119894+ 119871119901

119897119902119878119902

119894119896

(32)



119872119901

119894119896119897equiv 119879119901

119894119896119897+ 119885119901

119894119896119897 (33)

then we obtain

119872119901

119894119896119897= 119871119901

119897119894119896minus 119871119901

119897119896119894+ 119871119901

119897119902119878119902

119894119896+ 119871119901

119902119894119871119902

119897119896minus 119871119901

119902119896119871119902

119897119894 (34)


119877119901

119894119896119897= 119875119901

119894119896119897+119872119901

119894119896119897 (35)




119895119896


119877119901

119894119896119897+ 119877119901

119896119897119894+ 119877119901

119897119894119896= 119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896+ Γ119901

119902119896119878119902

119897119894+ Γ119901

119902119896119878119902

119894119896+ Γ119901

119902119894119878119902

119896119897

= 119878119901

119894119896119897+ Γ119902

119894119897119878119901

119902119896+ Γ119902

119896119897119878119901

119894119902+ 119878119901

119896119897119894+ Γ119902

119896119894119878119901

119902119897+ Γ119902

119897119894119878119901

119896119902

+ 119878119901

119897119894119896+ Γ119902

119897119896119878119901

119902119894+ Γ119902

119894119896119878119901

119897119902

= 119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896

+ 119878119901

119897119902119878119902

119894119896+ 119878119901

119896119902119878119902

119897119894+ 119878119901

119894119902119878119902

119896119897

(36)


119878119894

119895119901119878119901

119896119894+ 119878119894

119896119901119878119901

119894119895= 0

119878119894

119894119901119878119901

119895119896= 0

(37)


119878119894

119895119901119878119901

119896119894+ 119878119894

119896119901119878119901

119894119895+ 119878119894

119894119901119878119901

119895119896= 0 (38)




1198751199011015840

11989610158401198971015840= 119875119901

119896119897

1205971199091199011015840

120597119909119901

120597119909119896

1205971199091198961015840

120597119909119897

1205971199091198971015840+

1205972

119909119901

1205971199091198961015840

1205971199091198971015840

1205971199091199011015840

120597119909119901 (39)


119895119896equiv Γ119894

119895119896minus Γ119894

119896119895)




Value

119872119901

119896119897equiv

1

2

119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) (40)




1198892

119909119896

1198891205912= minusΓ119896

119894119895

119889119909119894

119889120591

119889119909119895

119889120591

(41)


119896119897+119872119901

119896119897 In (6) 120591


1198892

119909119896

1198891199042

= minusΓ119896

119894119895

119889119909119894

119889119904

119889119909119895

119889119904

(42)




119875119901

119896119897+119872119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894)

+

1

2

119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

(43)



1198892

119909119896

1198891199042

= minusΓ119896

119894119895

119889119909119894

119889119904

119889119909119895

119889119904

(44)



= 119909119894

(119905) 119905 isin [1199051 1199052] We


120575119878 = int

1199052

1199051

120575radic119892119894119895

119889119909119894

119889119905

119889119909119895

119889119905

119889119905

= int

1199052

1199051

120575 (119892119894119895(119889119909119894

119889119905) (119889119909119895

119889119905))

2radic119892119894119895(119889119909119894119889119905) (119889119909

119895119889119905)

119889119905

120575 (119892119894119895

119889119909119894

119889119905

119889119909119895

119889119905

) = 119892119894119895119863

119889119909119894

119889119905

119889119909119895

119889119905

+ 119892119894119895

119889119909119894

119889119905

119863

119889119909119895

119889119905

= 2119892119894119895

119889119909119894

119889119905

119863

119889119909119895

119889119905

119863

119889119909119895

119889119905

= 120575

119889119909119895

119889119905

+ Γ119895

119901119896

119889119909119901

119889119905

120575119909119896

119863

120575119909119895

119889119905

=

119889

119889119905

120575119909119895

+ Γ119895

119896119901

119889119909119896

119889119905

120575119909119901

119863

119889119909119895

119889119905

= 119863

120575119909119895

119889119905

+ 119878119895

119901119896

119889119909119901

119889119905

120575119909119896

(45)


120575(119892119894119895

119889119909119894

119889119905

119889119909119895

119889119905

) = 2119892119894119895

119889119909119894

119889119905

(119863

120575119909119895

119889119905

+ 119878119895

119901119896

119889119909119901

119889119905

120575119909119896

)

120575119904 = int

1199052

1199051

119892119894119895

119889119909119894

119889119904

119863120575119909119895

+ int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119904

119889119909119901

120575119909119896

= int

1199052

1199051

119863(119892119894119895

119889119909119894

119889119904

120575119909119895

)

minus int

1199052

1199051

119892119894119895119863

119889119909119894

119889119904

120575119909119895

+ int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119904

119889119909119901

120575119909119896

(46)


120575119904 = int

1199052

1199051

(119892119894119895119878119895

119901119896

119889119909119894

119889119905

119889119909119901

120575119909119896

minus 119892119894119895119863

119889119909119894

119889119905

120575119909119895

) (47)


int

1199052

1199051

(119892119894119895119878119895

119901119896

119889119909119894

119889119904

119889119909119901

120575119909119896

minus 119892119894119895119863

119889119909119894

119889119904

120575119909119895

) = 0 (48)


119892119894119896119878119896

119901119895

119889119909119894

119889119904

119889119909119901

minus 119892119894119895119863

119889119909119894

119889119904

= 0 (49)


119902

119892119895119902

119892119894119896119878119896

119901119895120585119894

119889119909119901


120575119904 = int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119905

119889119909119901

120575119909119896

(50)


119894119896and 119878

119895





119895



int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119905

119889119909119901

120575119909119896

(51)


119894119895


119896

= 119878119896

119894119895119860119894

119861119895


119894

120591 and 119861119895

120591


119901119902119878119901

119894119895119860119894

119861119895

119878119902

119896119897119860119896

119861119897

1205914


119894119896) with







119894119895


1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (52)

that is

Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (53)


Γ119896

119894119895=

1

2

119892119896119901

(119892119894119901119895119897

+ 119892119895119901119894

minus 119892119894119895119901

+ 119892119894119898119878119898

119895119901+ 119892119895119898119878119898

119894119901)

+

1

2

119878119896

119894119895

(54)



Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (55)



Γ119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

+ 119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

+

1

2

119878119901

119896119897

(56)


119875119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894) (57)


119901

119896119897+ 119872119901

119896119897 here 119872119896



119861119894

= 119886119860119894

(58)


119889119860119896

= minus (119875119896

119894119895+119872119896

119894119895)119860119894

119889119909119895

119889119861119896

= minus119875119896

119894119895119861119894

119889119909119895

(59)

Then we have

119860119896

119889119886 + 119886119889119860119896

= minus119875119896

119894119895119886119860119894

119889119909119895

(60)

119860119896

119889119886

119886

= 119872119896

119894119895119860119894

119889119909119895

(61)



119896119897+119872119901

119896119897


119889 ln 119886119889120591

119860119896

= 119872119896

119894119895119860119894

119860119895

(62)



119860119897

119872119896

119894119895119860119894

119860119895

minus 119860119896

119872119897

119894119895119860119894

119860119895

= 0 (63)

or

120575119897

119898119872119896

119894119895119860119894

119860119895

119860119898

minus 120575119896

119898119872119897

119894119895119860119894

119860119895

119860119898

= 0 (64)




120575119897

119898119872119896

119894119895minus 120575119896

119898119872119897

119894119895+ 120575119897

119894119872119896

119895119898minus 120575119896

119894119872119897

119895119898+ 120575119897

119895119872119896

119898119894minus 120575119896

119895119872119897

119898119894

= 0

(65)


have

119872119896

119894119895=

1

119899 + 1

(120575119896

119894119872119897

119895119897+ 120575119896

119895119872119897

119894119897) (66)



119894119895


119894119895 then let 119872119901

119896119897equiv

(12)119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) then119872119897

119896119897equiv (12)119878

119897

119896119897

We obtained

Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (67)



119894119895


119894119895=

(12)(1(119899 + 1))(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897



119896

119894119895119860119894

119889119909119895



119894119895







119894119895= 0 Assume that 119894 119872 sub 119875


119909rarr

119879119909119875|119872

rarr 119879perp



perp



119883119884 = 119879(nabla

1015840

119883119884) + perp (nabla

1015840

119883119884) Let it be

nabla119883119884 = 119879(nabla

1015840

119883119884) 120572(119883 119884) = perp (nabla




⟨1198771015840

(119883 119884)119885119882⟩ = ⟨119877 (119883 119884)119885119882⟩

+ ⟨120572 (119883 119885) 120572 (119884119882)⟩

minus ⟨120572 (119884 119885) 120572 (119883119882)⟩

(68)



(nabla10158401015840

119883120572) (119884 119885) = 119863

119883(120572 (119884 119885)) minus 120572 (nabla

119883119884119885)

minus 120572 (119884 nabla119883119885)

(69)

Codazzi equation is

perp ⟨1198771015840

(119883 119884)119885⟩ = (nabla10158401015840

119883120572) (119884 119885) minus (nabla

10158401015840

119884120572) (119883 119885) (70)





119909119894

= 119909119894

(1199101

119910119899minus1

) (71)


119886120572120573

= 119892119894119895

120597119909119894

120597119910120572

120597119909119895

120597119910120573

(72)


120572120573of



we assume that 119866120572120573120574



120572120573


119894119895and

119878119896

119894119895 we have

119866120572

120573120574

=

1

2

(119886120572120578

(119886120573120578120574

+ 119886120574120578120573

+ 119886120573120574120578

+ 119886120573120583119879120583

120574120578+ 119886120574120583119879120583

120573120578) + 119879120572

120574120573)

(73)


119866120572



1015840

by the formula

119866120572

120573120574= 1198661205721015840

12057310158401205741015840

120597119906120572

1205971199061205721015840

1205971199061205731015840

120597119906120573

1205971199061205741015840

120597119906120574+

120597119906120572

1205971199061205721015840

1205972

1199061205721015840

120597119906120573120597119906120574

(74)




119866120572

120573120574

120597119909119896

120597119906120572= Γ119896

119894119895

120597119909119894

120597119906120573

120597119909119895

120597119906120574+

1205972

119909119896

120597119906120573120597119906120574

(75)

We obtain

1

2

(119886120572120578

(119886120573120578120574

+ 119886120574120578120573

+ 119886120573120574120578

+ 119886120573120583119879120583

120574120578+ 119886120574120583119879120583

120573120578) + 119879120572

120574120573)

120597119909119896

120597119906120572

=

1

2

(119892119896119899

(119892119899119894119895

+ 119892119899119895119894

minus 119892119894119895119899

+ 119892119894119898119878119898

119895119899+ 119892119895119898119878119898

119894119899) + 119878119896

119894119895)

times

120597119909119894

120597119906120573

120597119909119895

120597119906120574+

1205972

119909119896

120597119906120573120597119906120574

(76)


119879120574

120572120573= 119886120574120578

119892119901119902119878119901

119894119895

120597119909119894

120597119910120572

120597119909119895

120597119910120573

120597119909119902

120597119910120578 (77)

using tensors 119886120572120573

and 119879120574












119860119894120572

119895120573120574= 119860119894120572

119895120573120574+ Γ119894

119901119896119860119901120572

119895120573

120597119909119896

120597119910120574minus Γ119901

119895119902119860119894120572

119901120573

120597119909119902

120597119910120574+ 119866120572

120578120574119860119894120578

119895120573

minus 119866120578

120573120574119860119894120572

119895120578

(78)


119906119894120572120573

minus 119906119894120573120572

= 119877119901

119896119897119894119906119901

120597119909119897

120597119910120572

120597119909119896

120597119910120573

+ 119878119902

119896119897119906119894119902

120597119909119897

120597119910120572

120597119909119896

120597119910120573

119906120574120572120573

minus 119906120574120573120572

= 119877120578

120573120572120574119906120578+ 119879120578

120573120572119906120574120578

(79)

where 119877120578



120573120574


120585119894

120572=

120597119909119894

120597119910120572 (80)



120585119894

1 120585

119894

119899minus1 ]119894 (81)

where 120585119894

1 120585

119894




120572

120585119894

120572120574= 120585119894

120572120574+ Γ119894

119901119902120585119901

120572

120597119909119902

120597119910120574minus 119866120578

120572120574120585119894

120578 (82)


120585119894

120572120574minus 120585119894

120574120572= 119878119894

119901119902120585119901

120572120585119902

120574+ 119879120578

120574120572120585119894

120578 (83)


0 = 119886120572120573120574

= (119892119894119895120585119894

120572120585119895

120573)

120574

= 119892119894119895120585119894

120572120574120585119895

120573+ 119892119894119895120585119894

120572120585119895

120573120574 (84)

we obtain

119892119894119895120585119894

120572120574120585119895

120573= 0 (85)


120585119894

120573120572= 120587120572120573]119894 (86)

Remark 11 120587120572120573



120574) the formula

119892119894119895]119894120574120585119894

120572= minus120587120574120572 (87)


]119894120574= minus119886120578120583

120587120583120574120585119894

120578 (88)



Further we obtain

120585119894

120573120594120582minus 120585119894

120573120582120594= minus119877119894

119896119897119901120585119896

120582120585119897

120594120585119901

120573+ 119877120590

120582120594120573120585119894

120590+ 119879120590

120582120594120585119894

120573120590

= (120587120594120573120582

minus 120587120582120573120594

) ]119894

minus (120587120594120573120587120578120582119886120578120590

minus 120587120582120573120587120578120594119886120578120590

) 120585119894

120590

(89)


120572 we have

119877120572120582120594120573

= 119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573120585119894

120572minus (120587120594120573120587120572120582

minus 120587120582120573120587120572120594) (90)


]119894120594120582

minus ]119894120582120594

= minus119877119894

119896119897119901120585119896

120582120585119897

120594]119901 + 119879

120590

120582120594]119894120590

= (120587120578120582120594

119886120578120590

minus 120587120578120594120582

119886120578120590

) 120585119894

120590

(91)


minus119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573]119894 + 119879

120590

120582120594120587120590120573

= 120587120594120573120582

minus 120587120582120573120594

(92)



minus119877119894119896119897119901

120585119896

120582120585119897

120594]119901120585119894120572+ 119879120590

120582120594120587120572120590

= 120587120572120582120594

minus 120587120572120594120582

(93)










119878119894

119895119896119901119902minus 119878119894

119895119896119902119901= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

+ 119878119905

119902119901119878119894

119895119896119905

(94)

or

119878119894

119895119896119901119902minus 119878119894

119895119896119902119901minus 119878119905

119902119901119878119894

119895119896119905

= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

(95)


119878119894

119895119896119901119894minus 119878119894

119895119896119894119901minus 119878119905

119894119901119878119894

119895119896119905

= (119878119894

119895119896119901minus 119878119894

119902119901119878119902

119895119896)

119894

minus 119878119894

119895119896119894119901minus 119878119894

119901119902119894119878119902

119895119896

(96)


(119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896)119894

minus 119892119896119901

119892119895119904

119878119894

119904119896119894119901minus 119892119896119901

119892119895119904

119878119894

119901119902119894119878119902

119904119896

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905

minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(97)


119867119895119894

= 119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896

119865119895119901

= 119892119896119901

119892119895119904

119878119894

119904119896119894

(98)


119867119895119894

119894minus 119865119895119894

119894minus 119892119896119901

119892119895119904

119878119902

119904119896119865119901119902

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(99)

where 119865119901119902

= 119878119894

119901119902119894


119894119896119897+ 119877119901

119896119897119894+ 119877119901

119897119894119896= 0 hence

119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896+ 119878119901

119897119902119878119902

119894119896+ 119878119901

119896119902119878119902

119897119894+ 119878119901

119894119902119878119902

119896119897= 0 (100)


119878119901

119894119896119901+ 119878119901

119896119901119894+ 119878119901

119901119894119896= 0 (101)




119894119895119878119902


expression

119878119901

119894119895119901= 120593119894119895minus 120593119895119894 (102)

Next if we put 119878119901119894119895119901


=



119894 System (99)

takes the form

119867119895119894

119894= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

119865119894119895

= 0

(103)


119862119894119895119896

= 119892119901119895

119892119902119896

119878119894

119901119902+ 119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896

119901119902 (104)


119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+ 119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904

minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

(105)



119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

=

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

(106)

hence

119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902) (107)


119862119894119896119895

119894= minus119862119894119895119896

119894= minus (119862

119894119895119896

119894+ Γ119895

119901119894119862119894119901119896

+ Γ119896

119901119894119862119894119895119901

+ Γ119894

119901119894119862119901119896119895

) (108)


= 119892119901119895

119892119902119896

119878119894

119901119902+

119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896


Γ119895

119901119894119862119894119901119896

= Γ119895

119894119901119862119894119901119896

=

1

2

(Γ119895

119894119901119862119894119901119896

+ Γ119895

119901119894119862119901119894119896

) =

1

2

119878119895

119894119901119862119901119896119894

= minus

1

2

119878119895

119901119894119862119896119901119894

(109)

similarly we obtain

Γ119896

119901119894119862119894119895119901

= Γ119896

119894119901119862119901119895119894

=

1

2

119862119895119901119894

(Γ119896

119894119901minus Γ119896

119901119894) =

1

2

119878119896

119894119901119862119895119901119894

=

1

2

119878119896

119901119902119862119895119902119901

(110)

Then we write

119862119894119896119895

119894= minus119862119894119895119896

119894

= minus119862119894119895119896

119894minus

1

2

119878119895

119901119902119862119896119901119902

+

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894+

1

2

119878119895

119901119902119862119896119901119902

minus

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

(111)



119901

119901119897=

(12)119892119894119901119897119892119894119901

= (1radic119892)(120597radic119892120597119909119897

) and Γ119901

119897119901= Γ119901

119901119897+ 119878119901

119897119901and

obtain

Γ119901

119901119897=

1

radicminus119892


120597119909119897

+ (ln120595)119897= (ln (120595radicminus119892))

119897 (112)

Then we obtain

119867119895119896

minus 119867119896119895

minus 119865119895119896

= minus119862119894119895119896


119894119862119894119896119895

(113)


120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


119894+ (ln (120595radicminus119892))

119894119862119894119896119895

)

120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


)119894

(114)


(120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

))119896

= 0 (115)







119894

119895119896 we



119894

119895119896and 119892

119894119896


119877119894119895= Γ119901

119894119901119895minus Γ119901

119894119895119901+ Γ119901

119902119895Γ119902

119894119901minus Γ119901

119902119901Γ119902

119894119895 (116)


119894119897119878119897

119896119895)119892119894119896


Γ119894

119895119896minus Γ119894


int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881 (117)




119877119894119896+ 119878119895

119894119897119878119897

119896119895= 0 (118)


119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901

minus 120575119895

119896(119892119894119901

119901minus

1

2

119892119894119901

119892119898119899119892119898119899

119901+ 119892119901119902

Γ119894

119901119902) minus 119892119894119895

Γ119901

119896119901

+ 2 (119892119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(119)



minus3119892119894119895

119895+

3

2

119892119894119895

119892119901119902119892119901119902

119895minus 3119892119901119902

Γ119894

119901119902+ 119892119894119901

119878119895

119901119895= 0 (120)

Therefore we have

119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901minus 119892119894119895

Γ119901

119896119901

minus

1

3

120575119895

119896119892119894119901

119878119902

119901119902+ 2 (119892

119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(121)


minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901+ 2 (119892

119901119895119878119901

119894119896+ 119892119894119901119878119901

119896119895) = 0

(122)



5 (119892119894119901119878119901

119896119895+ 119892119895119901119878119901

119894119896) +

1

3

(119892119895119896119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0 (123)

Further we obtained

minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901minus

2

15

(119892119896119895119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0

(124)


0 then from (123) we have 119878119894




119898+

119882119892) = 0 where 119882

119898and 119882




120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int (119877119894119896radicminus119892120575119892

119894119896

+ 119877119894119896119892119894119896

120575radicminus119892 + 119892119894119896

radicminus119892120575119877119894119896

+ 119878119895

119894119897119878119897

119896119895radicminus119892120575119892

119894119896

+ 119878119895

119894119897119878119897

119896119895119892119894119896

120575radicminus119892) 119889119881

119877119894119896119892119894119896


1

2

119877119901119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(125)

Similarly we obtain

119878119895

119894119897119878119897

119896119895119892119894119896


1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(126)



119894119896

(120575Γ119897

119896119894)119897minus 119892119894119896

(120575Γ119897

119896119897)119894= (119892119894119896

120575Γ119897

119896119894minus 119892119894119897

120575Γ119901

119897119901)119897



119894119896= 119892119894119896

120575(Γ119901

119902119901Γ119902

119896119894minus

Γ119901

119902119894Γ119902119897



119892119894119896

120575119877119894119896= (119878119901

119894119901119878119902

119902119896minus 119878119901

119894119902119878119902

119901119896minus 119892119896119901119892119902119905

119878119898

119894119905119878119901

119902119898) 120575119892119894119896

(127)

Thus we have

120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(128)


120575119882119892

= 1198701int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= 1198701int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(129)


1198883


120575119882119892= 1198702int119879119894119896radicminus119892120575119892

119894119896

119889119881 (130)


2usually


119892+120575119882119898=

0 we find relations

int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902


119894119896

119889119881 = 0

(131)



119877119894119896minus

1

2

119892119894119896119877 + 119878119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902+ 119878119895

119894119897119878119897

119896119895

minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896= 119870119879119894119896

(132)


2

7 Conclusions







References




























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






Value

119872119901

119896119897equiv

1

2

119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) (40)




1198892

119909119896

1198891205912= minusΓ119896

119894119895

119889119909119894

119889120591

119889119909119895

119889120591

(41)


119896119897+119872119901

119896119897 In (6) 120591


1198892

119909119896

1198891199042

= minusΓ119896

119894119895

119889119909119894

119889119904

119889119909119895

119889119904

(42)




119875119901

119896119897+119872119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894)

+

1

2

119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

(43)



1198892

119909119896

1198891199042

= minusΓ119896

119894119895

119889119909119894

119889119904

119889119909119895

119889119904

(44)



= 119909119894

(119905) 119905 isin [1199051 1199052] We


120575119878 = int

1199052

1199051

120575radic119892119894119895

119889119909119894

119889119905

119889119909119895

119889119905

119889119905

= int

1199052

1199051

120575 (119892119894119895(119889119909119894

119889119905) (119889119909119895

119889119905))

2radic119892119894119895(119889119909119894119889119905) (119889119909

119895119889119905)

119889119905

120575 (119892119894119895

119889119909119894

119889119905

119889119909119895

119889119905

) = 119892119894119895119863

119889119909119894

119889119905

119889119909119895

119889119905

+ 119892119894119895

119889119909119894

119889119905

119863

119889119909119895

119889119905

= 2119892119894119895

119889119909119894

119889119905

119863

119889119909119895

119889119905

119863

119889119909119895

119889119905

= 120575

119889119909119895

119889119905

+ Γ119895

119901119896

119889119909119901

119889119905

120575119909119896

119863

120575119909119895

119889119905

=

119889

119889119905

120575119909119895

+ Γ119895

119896119901

119889119909119896

119889119905

120575119909119901

119863

119889119909119895

119889119905

= 119863

120575119909119895

119889119905

+ 119878119895

119901119896

119889119909119901

119889119905

120575119909119896

(45)


120575(119892119894119895

119889119909119894

119889119905

119889119909119895

119889119905

) = 2119892119894119895

119889119909119894

119889119905

(119863

120575119909119895

119889119905

+ 119878119895

119901119896

119889119909119901

119889119905

120575119909119896

)

120575119904 = int

1199052

1199051

119892119894119895

119889119909119894

119889119904

119863120575119909119895

+ int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119904

119889119909119901

120575119909119896

= int

1199052

1199051

119863(119892119894119895

119889119909119894

119889119904

120575119909119895

)

minus int

1199052

1199051

119892119894119895119863

119889119909119894

119889119904

120575119909119895

+ int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119904

119889119909119901

120575119909119896

(46)


120575119904 = int

1199052

1199051

(119892119894119895119878119895

119901119896

119889119909119894

119889119905

119889119909119901

120575119909119896

minus 119892119894119895119863

119889119909119894

119889119905

120575119909119895

) (47)


int

1199052

1199051

(119892119894119895119878119895

119901119896

119889119909119894

119889119904

119889119909119901

120575119909119896

minus 119892119894119895119863

119889119909119894

119889119904

120575119909119895

) = 0 (48)


119892119894119896119878119896

119901119895

119889119909119894

119889119904

119889119909119901

minus 119892119894119895119863

119889119909119894

119889119904

= 0 (49)


119902

119892119895119902

119892119894119896119878119896

119901119895120585119894

119889119909119901


120575119904 = int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119905

119889119909119901

120575119909119896

(50)


119894119896and 119878

119895





119895



int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119905

119889119909119901

120575119909119896

(51)


119894119895


119896

= 119878119896

119894119895119860119894

119861119895


119894

120591 and 119861119895

120591


119901119902119878119901

119894119895119860119894

119861119895

119878119902

119896119897119860119896

119861119897

1205914


119894119896) with







119894119895


1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (52)

that is

Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (53)


Γ119896

119894119895=

1

2

119892119896119901

(119892119894119901119895119897

+ 119892119895119901119894

minus 119892119894119895119901

+ 119892119894119898119878119898

119895119901+ 119892119895119898119878119898

119894119901)

+

1

2

119878119896

119894119895

(54)



Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (55)



Γ119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

+ 119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

+

1

2

119878119901

119896119897

(56)


119875119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894) (57)


119901

119896119897+ 119872119901

119896119897 here 119872119896



119861119894

= 119886119860119894

(58)


119889119860119896

= minus (119875119896

119894119895+119872119896

119894119895)119860119894

119889119909119895

119889119861119896

= minus119875119896

119894119895119861119894

119889119909119895

(59)

Then we have

119860119896

119889119886 + 119886119889119860119896

= minus119875119896

119894119895119886119860119894

119889119909119895

(60)

119860119896

119889119886

119886

= 119872119896

119894119895119860119894

119889119909119895

(61)



119896119897+119872119901

119896119897


119889 ln 119886119889120591

119860119896

= 119872119896

119894119895119860119894

119860119895

(62)



119860119897

119872119896

119894119895119860119894

119860119895

minus 119860119896

119872119897

119894119895119860119894

119860119895

= 0 (63)

or

120575119897

119898119872119896

119894119895119860119894

119860119895

119860119898

minus 120575119896

119898119872119897

119894119895119860119894

119860119895

119860119898

= 0 (64)




120575119897

119898119872119896

119894119895minus 120575119896

119898119872119897

119894119895+ 120575119897

119894119872119896

119895119898minus 120575119896

119894119872119897

119895119898+ 120575119897

119895119872119896

119898119894minus 120575119896

119895119872119897

119898119894

= 0

(65)


have

119872119896

119894119895=

1

119899 + 1

(120575119896

119894119872119897

119895119897+ 120575119896

119895119872119897

119894119897) (66)



119894119895


119894119895 then let 119872119901

119896119897equiv

(12)119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) then119872119897

119896119897equiv (12)119878

119897

119896119897

We obtained

Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (67)



119894119895


119894119895=

(12)(1(119899 + 1))(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897



119896

119894119895119860119894

119889119909119895



119894119895







119894119895= 0 Assume that 119894 119872 sub 119875


119909rarr

119879119909119875|119872

rarr 119879perp



perp



119883119884 = 119879(nabla

1015840

119883119884) + perp (nabla

1015840

119883119884) Let it be

nabla119883119884 = 119879(nabla

1015840

119883119884) 120572(119883 119884) = perp (nabla




⟨1198771015840

(119883 119884)119885119882⟩ = ⟨119877 (119883 119884)119885119882⟩

+ ⟨120572 (119883 119885) 120572 (119884119882)⟩

minus ⟨120572 (119884 119885) 120572 (119883119882)⟩

(68)



(nabla10158401015840

119883120572) (119884 119885) = 119863

119883(120572 (119884 119885)) minus 120572 (nabla

119883119884119885)

minus 120572 (119884 nabla119883119885)

(69)

Codazzi equation is

perp ⟨1198771015840

(119883 119884)119885⟩ = (nabla10158401015840

119883120572) (119884 119885) minus (nabla

10158401015840

119884120572) (119883 119885) (70)





119909119894

= 119909119894

(1199101

119910119899minus1

) (71)


119886120572120573

= 119892119894119895

120597119909119894

120597119910120572

120597119909119895

120597119910120573

(72)


120572120573of



we assume that 119866120572120573120574



120572120573


119894119895and

119878119896

119894119895 we have

119866120572

120573120574

=

1

2

(119886120572120578

(119886120573120578120574

+ 119886120574120578120573

+ 119886120573120574120578

+ 119886120573120583119879120583

120574120578+ 119886120574120583119879120583

120573120578) + 119879120572

120574120573)

(73)


119866120572



1015840

by the formula

119866120572

120573120574= 1198661205721015840

12057310158401205741015840

120597119906120572

1205971199061205721015840

1205971199061205731015840

120597119906120573

1205971199061205741015840

120597119906120574+

120597119906120572

1205971199061205721015840

1205972

1199061205721015840

120597119906120573120597119906120574

(74)




119866120572

120573120574

120597119909119896

120597119906120572= Γ119896

119894119895

120597119909119894

120597119906120573

120597119909119895

120597119906120574+

1205972

119909119896

120597119906120573120597119906120574

(75)

We obtain

1

2

(119886120572120578

(119886120573120578120574

+ 119886120574120578120573

+ 119886120573120574120578

+ 119886120573120583119879120583

120574120578+ 119886120574120583119879120583

120573120578) + 119879120572

120574120573)

120597119909119896

120597119906120572

=

1

2

(119892119896119899

(119892119899119894119895

+ 119892119899119895119894

minus 119892119894119895119899

+ 119892119894119898119878119898

119895119899+ 119892119895119898119878119898

119894119899) + 119878119896

119894119895)

times

120597119909119894

120597119906120573

120597119909119895

120597119906120574+

1205972

119909119896

120597119906120573120597119906120574

(76)


119879120574

120572120573= 119886120574120578

119892119901119902119878119901

119894119895

120597119909119894

120597119910120572

120597119909119895

120597119910120573

120597119909119902

120597119910120578 (77)

using tensors 119886120572120573

and 119879120574












119860119894120572

119895120573120574= 119860119894120572

119895120573120574+ Γ119894

119901119896119860119901120572

119895120573

120597119909119896

120597119910120574minus Γ119901

119895119902119860119894120572

119901120573

120597119909119902

120597119910120574+ 119866120572

120578120574119860119894120578

119895120573

minus 119866120578

120573120574119860119894120572

119895120578

(78)


119906119894120572120573

minus 119906119894120573120572

= 119877119901

119896119897119894119906119901

120597119909119897

120597119910120572

120597119909119896

120597119910120573

+ 119878119902

119896119897119906119894119902

120597119909119897

120597119910120572

120597119909119896

120597119910120573

119906120574120572120573

minus 119906120574120573120572

= 119877120578

120573120572120574119906120578+ 119879120578

120573120572119906120574120578

(79)

where 119877120578



120573120574


120585119894

120572=

120597119909119894

120597119910120572 (80)



120585119894

1 120585

119894

119899minus1 ]119894 (81)

where 120585119894

1 120585

119894




120572

120585119894

120572120574= 120585119894

120572120574+ Γ119894

119901119902120585119901

120572

120597119909119902

120597119910120574minus 119866120578

120572120574120585119894

120578 (82)


120585119894

120572120574minus 120585119894

120574120572= 119878119894

119901119902120585119901

120572120585119902

120574+ 119879120578

120574120572120585119894

120578 (83)


0 = 119886120572120573120574

= (119892119894119895120585119894

120572120585119895

120573)

120574

= 119892119894119895120585119894

120572120574120585119895

120573+ 119892119894119895120585119894

120572120585119895

120573120574 (84)

we obtain

119892119894119895120585119894

120572120574120585119895

120573= 0 (85)


120585119894

120573120572= 120587120572120573]119894 (86)

Remark 11 120587120572120573



120574) the formula

119892119894119895]119894120574120585119894

120572= minus120587120574120572 (87)


]119894120574= minus119886120578120583

120587120583120574120585119894

120578 (88)



Further we obtain

120585119894

120573120594120582minus 120585119894

120573120582120594= minus119877119894

119896119897119901120585119896

120582120585119897

120594120585119901

120573+ 119877120590

120582120594120573120585119894

120590+ 119879120590

120582120594120585119894

120573120590

= (120587120594120573120582

minus 120587120582120573120594

) ]119894

minus (120587120594120573120587120578120582119886120578120590

minus 120587120582120573120587120578120594119886120578120590

) 120585119894

120590

(89)


120572 we have

119877120572120582120594120573

= 119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573120585119894

120572minus (120587120594120573120587120572120582

minus 120587120582120573120587120572120594) (90)


]119894120594120582

minus ]119894120582120594

= minus119877119894

119896119897119901120585119896

120582120585119897

120594]119901 + 119879

120590

120582120594]119894120590

= (120587120578120582120594

119886120578120590

minus 120587120578120594120582

119886120578120590

) 120585119894

120590

(91)


minus119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573]119894 + 119879

120590

120582120594120587120590120573

= 120587120594120573120582

minus 120587120582120573120594

(92)



minus119877119894119896119897119901

120585119896

120582120585119897

120594]119901120585119894120572+ 119879120590

120582120594120587120572120590

= 120587120572120582120594

minus 120587120572120594120582

(93)










119878119894

119895119896119901119902minus 119878119894

119895119896119902119901= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

+ 119878119905

119902119901119878119894

119895119896119905

(94)

or

119878119894

119895119896119901119902minus 119878119894

119895119896119902119901minus 119878119905

119902119901119878119894

119895119896119905

= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

(95)


119878119894

119895119896119901119894minus 119878119894

119895119896119894119901minus 119878119905

119894119901119878119894

119895119896119905

= (119878119894

119895119896119901minus 119878119894

119902119901119878119902

119895119896)

119894

minus 119878119894

119895119896119894119901minus 119878119894

119901119902119894119878119902

119895119896

(96)


(119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896)119894

minus 119892119896119901

119892119895119904

119878119894

119904119896119894119901minus 119892119896119901

119892119895119904

119878119894

119901119902119894119878119902

119904119896

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905

minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(97)


119867119895119894

= 119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896

119865119895119901

= 119892119896119901

119892119895119904

119878119894

119904119896119894

(98)


119867119895119894

119894minus 119865119895119894

119894minus 119892119896119901

119892119895119904

119878119902

119904119896119865119901119902

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(99)

where 119865119901119902

= 119878119894

119901119902119894


119894119896119897+ 119877119901

119896119897119894+ 119877119901

119897119894119896= 0 hence

119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896+ 119878119901

119897119902119878119902

119894119896+ 119878119901

119896119902119878119902

119897119894+ 119878119901

119894119902119878119902

119896119897= 0 (100)


119878119901

119894119896119901+ 119878119901

119896119901119894+ 119878119901

119901119894119896= 0 (101)




119894119895119878119902


expression

119878119901

119894119895119901= 120593119894119895minus 120593119895119894 (102)

Next if we put 119878119901119894119895119901


=



119894 System (99)

takes the form

119867119895119894

119894= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

119865119894119895

= 0

(103)


119862119894119895119896

= 119892119901119895

119892119902119896

119878119894

119901119902+ 119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896

119901119902 (104)


119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+ 119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904

minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

(105)



119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

=

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

(106)

hence

119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902) (107)


119862119894119896119895

119894= minus119862119894119895119896

119894= minus (119862

119894119895119896

119894+ Γ119895

119901119894119862119894119901119896

+ Γ119896

119901119894119862119894119895119901

+ Γ119894

119901119894119862119901119896119895

) (108)


= 119892119901119895

119892119902119896

119878119894

119901119902+

119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896


Γ119895

119901119894119862119894119901119896

= Γ119895

119894119901119862119894119901119896

=

1

2

(Γ119895

119894119901119862119894119901119896

+ Γ119895

119901119894119862119901119894119896

) =

1

2

119878119895

119894119901119862119901119896119894

= minus

1

2

119878119895

119901119894119862119896119901119894

(109)

similarly we obtain

Γ119896

119901119894119862119894119895119901

= Γ119896

119894119901119862119901119895119894

=

1

2

119862119895119901119894

(Γ119896

119894119901minus Γ119896

119901119894) =

1

2

119878119896

119894119901119862119895119901119894

=

1

2

119878119896

119901119902119862119895119902119901

(110)

Then we write

119862119894119896119895

119894= minus119862119894119895119896

119894

= minus119862119894119895119896

119894minus

1

2

119878119895

119901119902119862119896119901119902

+

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894+

1

2

119878119895

119901119902119862119896119901119902

minus

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

(111)



119901

119901119897=

(12)119892119894119901119897119892119894119901

= (1radic119892)(120597radic119892120597119909119897

) and Γ119901

119897119901= Γ119901

119901119897+ 119878119901

119897119901and

obtain

Γ119901

119901119897=

1

radicminus119892


120597119909119897

+ (ln120595)119897= (ln (120595radicminus119892))

119897 (112)

Then we obtain

119867119895119896

minus 119867119896119895

minus 119865119895119896

= minus119862119894119895119896


119894119862119894119896119895

(113)


120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


119894+ (ln (120595radicminus119892))

119894119862119894119896119895

)

120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


)119894

(114)


(120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

))119896

= 0 (115)







119894

119895119896 we



119894

119895119896and 119892

119894119896


119877119894119895= Γ119901

119894119901119895minus Γ119901

119894119895119901+ Γ119901

119902119895Γ119902

119894119901minus Γ119901

119902119901Γ119902

119894119895 (116)


119894119897119878119897

119896119895)119892119894119896


Γ119894

119895119896minus Γ119894


int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881 (117)




119877119894119896+ 119878119895

119894119897119878119897

119896119895= 0 (118)


119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901

minus 120575119895

119896(119892119894119901

119901minus

1

2

119892119894119901

119892119898119899119892119898119899

119901+ 119892119901119902

Γ119894

119901119902) minus 119892119894119895

Γ119901

119896119901

+ 2 (119892119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(119)



minus3119892119894119895

119895+

3

2

119892119894119895

119892119901119902119892119901119902

119895minus 3119892119901119902

Γ119894

119901119902+ 119892119894119901

119878119895

119901119895= 0 (120)

Therefore we have

119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901minus 119892119894119895

Γ119901

119896119901

minus

1

3

120575119895

119896119892119894119901

119878119902

119901119902+ 2 (119892

119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(121)


minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901+ 2 (119892

119901119895119878119901

119894119896+ 119892119894119901119878119901

119896119895) = 0

(122)



5 (119892119894119901119878119901

119896119895+ 119892119895119901119878119901

119894119896) +

1

3

(119892119895119896119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0 (123)

Further we obtained

minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901minus

2

15

(119892119896119895119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0

(124)


0 then from (123) we have 119878119894




119898+

119882119892) = 0 where 119882

119898and 119882




120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int (119877119894119896radicminus119892120575119892

119894119896

+ 119877119894119896119892119894119896

120575radicminus119892 + 119892119894119896

radicminus119892120575119877119894119896

+ 119878119895

119894119897119878119897

119896119895radicminus119892120575119892

119894119896

+ 119878119895

119894119897119878119897

119896119895119892119894119896

120575radicminus119892) 119889119881

119877119894119896119892119894119896


1

2

119877119901119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(125)

Similarly we obtain

119878119895

119894119897119878119897

119896119895119892119894119896


1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(126)



119894119896

(120575Γ119897

119896119894)119897minus 119892119894119896

(120575Γ119897

119896119897)119894= (119892119894119896

120575Γ119897

119896119894minus 119892119894119897

120575Γ119901

119897119901)119897



119894119896= 119892119894119896

120575(Γ119901

119902119901Γ119902

119896119894minus

Γ119901

119902119894Γ119902119897



119892119894119896

120575119877119894119896= (119878119901

119894119901119878119902

119902119896minus 119878119901

119894119902119878119902

119901119896minus 119892119896119901119892119902119905

119878119898

119894119905119878119901

119902119898) 120575119892119894119896

(127)

Thus we have

120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(128)


120575119882119892

= 1198701int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= 1198701int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(129)


1198883


120575119882119892= 1198702int119879119894119896radicminus119892120575119892

119894119896

119889119881 (130)


2usually


119892+120575119882119898=

0 we find relations

int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902


119894119896

119889119881 = 0

(131)



119877119894119896minus

1

2

119892119894119896119877 + 119878119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902+ 119878119895

119894119897119878119897

119896119895

minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896= 119870119879119894119896

(132)


2

7 Conclusions







References




























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119895



int

1199052

1199051

119892119894119895119878119895

119901119896

119889119909119894

119889119905

119889119909119901

120575119909119896

(51)


119894119895


119896

= 119878119896

119894119895119860119894

119861119895


119894

120591 and 119861119895

120591


119901119902119878119901

119894119895119860119894

119861119895

119878119902

119896119897119860119896

119861119897

1205914


119894119896) with







119894119895


1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (52)

that is

Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (53)


Γ119896

119894119895=

1

2

119892119896119901

(119892119894119901119895119897

+ 119892119895119901119894

minus 119892119894119895119901

+ 119892119894119898119878119898

119895119901+ 119892119895119898119878119898

119894119901)

+

1

2

119878119896

119894119895

(54)



Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (55)



Γ119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894

+ 119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894)

+

1

2

119878119901

119896119897

(56)


119875119901

119896119897=

1

2

119892119901119894

(119892119894119896119897

+ 119892119897119894119896

minus 119892119896119897119894) (57)


119901

119896119897+ 119872119901

119896119897 here 119872119896



119861119894

= 119886119860119894

(58)


119889119860119896

= minus (119875119896

119894119895+119872119896

119894119895)119860119894

119889119909119895

119889119861119896

= minus119875119896

119894119895119861119894

119889119909119895

(59)

Then we have

119860119896

119889119886 + 119886119889119860119896

= minus119875119896

119894119895119886119860119894

119889119909119895

(60)

119860119896

119889119886

119886

= 119872119896

119894119895119860119894

119889119909119895

(61)



119896119897+119872119901

119896119897


119889 ln 119886119889120591

119860119896

= 119872119896

119894119895119860119894

119860119895

(62)



119860119897

119872119896

119894119895119860119894

119860119895

minus 119860119896

119872119897

119894119895119860119894

119860119895

= 0 (63)

or

120575119897

119898119872119896

119894119895119860119894

119860119895

119860119898

minus 120575119896

119898119872119897

119894119895119860119894

119860119895

119860119898

= 0 (64)




120575119897

119898119872119896

119894119895minus 120575119896

119898119872119897

119894119895+ 120575119897

119894119872119896

119895119898minus 120575119896

119894119872119897

119895119898+ 120575119897

119895119872119896

119898119894minus 120575119896

119895119872119897

119898119894

= 0

(65)


have

119872119896

119894119895=

1

119899 + 1

(120575119896

119894119872119897

119895119897+ 120575119896

119895119872119897

119894119897) (66)



119894119895


119894119895 then let 119872119901

119896119897equiv

(12)119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) then119872119897

119896119897equiv (12)119878

119897

119896119897

We obtained

Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (67)



119894119895


119894119895=

(12)(1(119899 + 1))(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897



119896

119894119895119860119894

119889119909119895



119894119895







119894119895= 0 Assume that 119894 119872 sub 119875


119909rarr

119879119909119875|119872

rarr 119879perp



perp



119883119884 = 119879(nabla

1015840

119883119884) + perp (nabla

1015840

119883119884) Let it be

nabla119883119884 = 119879(nabla

1015840

119883119884) 120572(119883 119884) = perp (nabla




⟨1198771015840

(119883 119884)119885119882⟩ = ⟨119877 (119883 119884)119885119882⟩

+ ⟨120572 (119883 119885) 120572 (119884119882)⟩

minus ⟨120572 (119884 119885) 120572 (119883119882)⟩

(68)



(nabla10158401015840

119883120572) (119884 119885) = 119863

119883(120572 (119884 119885)) minus 120572 (nabla

119883119884119885)

minus 120572 (119884 nabla119883119885)

(69)

Codazzi equation is

perp ⟨1198771015840

(119883 119884)119885⟩ = (nabla10158401015840

119883120572) (119884 119885) minus (nabla

10158401015840

119884120572) (119883 119885) (70)





119909119894

= 119909119894

(1199101

119910119899minus1

) (71)


119886120572120573

= 119892119894119895

120597119909119894

120597119910120572

120597119909119895

120597119910120573

(72)


120572120573of



we assume that 119866120572120573120574



120572120573


119894119895and

119878119896

119894119895 we have

119866120572

120573120574

=

1

2

(119886120572120578

(119886120573120578120574

+ 119886120574120578120573

+ 119886120573120574120578

+ 119886120573120583119879120583

120574120578+ 119886120574120583119879120583

120573120578) + 119879120572

120574120573)

(73)


119866120572



1015840

by the formula

119866120572

120573120574= 1198661205721015840

12057310158401205741015840

120597119906120572

1205971199061205721015840

1205971199061205731015840

120597119906120573

1205971199061205741015840

120597119906120574+

120597119906120572

1205971199061205721015840

1205972

1199061205721015840

120597119906120573120597119906120574

(74)




119866120572

120573120574

120597119909119896

120597119906120572= Γ119896

119894119895

120597119909119894

120597119906120573

120597119909119895

120597119906120574+

1205972

119909119896

120597119906120573120597119906120574

(75)

We obtain

1

2

(119886120572120578

(119886120573120578120574

+ 119886120574120578120573

+ 119886120573120574120578

+ 119886120573120583119879120583

120574120578+ 119886120574120583119879120583

120573120578) + 119879120572

120574120573)

120597119909119896

120597119906120572

=

1

2

(119892119896119899

(119892119899119894119895

+ 119892119899119895119894

minus 119892119894119895119899

+ 119892119894119898119878119898

119895119899+ 119892119895119898119878119898

119894119899) + 119878119896

119894119895)

times

120597119909119894

120597119906120573

120597119909119895

120597119906120574+

1205972

119909119896

120597119906120573120597119906120574

(76)


119879120574

120572120573= 119886120574120578

119892119901119902119878119901

119894119895

120597119909119894

120597119910120572

120597119909119895

120597119910120573

120597119909119902

120597119910120578 (77)

using tensors 119886120572120573

and 119879120574












119860119894120572

119895120573120574= 119860119894120572

119895120573120574+ Γ119894

119901119896119860119901120572

119895120573

120597119909119896

120597119910120574minus Γ119901

119895119902119860119894120572

119901120573

120597119909119902

120597119910120574+ 119866120572

120578120574119860119894120578

119895120573

minus 119866120578

120573120574119860119894120572

119895120578

(78)


119906119894120572120573

minus 119906119894120573120572

= 119877119901

119896119897119894119906119901

120597119909119897

120597119910120572

120597119909119896

120597119910120573

+ 119878119902

119896119897119906119894119902

120597119909119897

120597119910120572

120597119909119896

120597119910120573

119906120574120572120573

minus 119906120574120573120572

= 119877120578

120573120572120574119906120578+ 119879120578

120573120572119906120574120578

(79)

where 119877120578



120573120574


120585119894

120572=

120597119909119894

120597119910120572 (80)



120585119894

1 120585

119894

119899minus1 ]119894 (81)

where 120585119894

1 120585

119894




120572

120585119894

120572120574= 120585119894

120572120574+ Γ119894

119901119902120585119901

120572

120597119909119902

120597119910120574minus 119866120578

120572120574120585119894

120578 (82)


120585119894

120572120574minus 120585119894

120574120572= 119878119894

119901119902120585119901

120572120585119902

120574+ 119879120578

120574120572120585119894

120578 (83)


0 = 119886120572120573120574

= (119892119894119895120585119894

120572120585119895

120573)

120574

= 119892119894119895120585119894

120572120574120585119895

120573+ 119892119894119895120585119894

120572120585119895

120573120574 (84)

we obtain

119892119894119895120585119894

120572120574120585119895

120573= 0 (85)


120585119894

120573120572= 120587120572120573]119894 (86)

Remark 11 120587120572120573



120574) the formula

119892119894119895]119894120574120585119894

120572= minus120587120574120572 (87)


]119894120574= minus119886120578120583

120587120583120574120585119894

120578 (88)



Further we obtain

120585119894

120573120594120582minus 120585119894

120573120582120594= minus119877119894

119896119897119901120585119896

120582120585119897

120594120585119901

120573+ 119877120590

120582120594120573120585119894

120590+ 119879120590

120582120594120585119894

120573120590

= (120587120594120573120582

minus 120587120582120573120594

) ]119894

minus (120587120594120573120587120578120582119886120578120590

minus 120587120582120573120587120578120594119886120578120590

) 120585119894

120590

(89)


120572 we have

119877120572120582120594120573

= 119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573120585119894

120572minus (120587120594120573120587120572120582

minus 120587120582120573120587120572120594) (90)


]119894120594120582

minus ]119894120582120594

= minus119877119894

119896119897119901120585119896

120582120585119897

120594]119901 + 119879

120590

120582120594]119894120590

= (120587120578120582120594

119886120578120590

minus 120587120578120594120582

119886120578120590

) 120585119894

120590

(91)


minus119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573]119894 + 119879

120590

120582120594120587120590120573

= 120587120594120573120582

minus 120587120582120573120594

(92)



minus119877119894119896119897119901

120585119896

120582120585119897

120594]119901120585119894120572+ 119879120590

120582120594120587120572120590

= 120587120572120582120594

minus 120587120572120594120582

(93)










119878119894

119895119896119901119902minus 119878119894

119895119896119902119901= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

+ 119878119905

119902119901119878119894

119895119896119905

(94)

or

119878119894

119895119896119901119902minus 119878119894

119895119896119902119901minus 119878119905

119902119901119878119894

119895119896119905

= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

(95)


119878119894

119895119896119901119894minus 119878119894

119895119896119894119901minus 119878119905

119894119901119878119894

119895119896119905

= (119878119894

119895119896119901minus 119878119894

119902119901119878119902

119895119896)

119894

minus 119878119894

119895119896119894119901minus 119878119894

119901119902119894119878119902

119895119896

(96)


(119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896)119894

minus 119892119896119901

119892119895119904

119878119894

119904119896119894119901minus 119892119896119901

119892119895119904

119878119894

119901119902119894119878119902

119904119896

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905

minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(97)


119867119895119894

= 119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896

119865119895119901

= 119892119896119901

119892119895119904

119878119894

119904119896119894

(98)


119867119895119894

119894minus 119865119895119894

119894minus 119892119896119901

119892119895119904

119878119902

119904119896119865119901119902

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(99)

where 119865119901119902

= 119878119894

119901119902119894


119894119896119897+ 119877119901

119896119897119894+ 119877119901

119897119894119896= 0 hence

119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896+ 119878119901

119897119902119878119902

119894119896+ 119878119901

119896119902119878119902

119897119894+ 119878119901

119894119902119878119902

119896119897= 0 (100)


119878119901

119894119896119901+ 119878119901

119896119901119894+ 119878119901

119901119894119896= 0 (101)




119894119895119878119902


expression

119878119901

119894119895119901= 120593119894119895minus 120593119895119894 (102)

Next if we put 119878119901119894119895119901


=



119894 System (99)

takes the form

119867119895119894

119894= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

119865119894119895

= 0

(103)


119862119894119895119896

= 119892119901119895

119892119902119896

119878119894

119901119902+ 119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896

119901119902 (104)


119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+ 119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904

minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

(105)



119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

=

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

(106)

hence

119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902) (107)


119862119894119896119895

119894= minus119862119894119895119896

119894= minus (119862

119894119895119896

119894+ Γ119895

119901119894119862119894119901119896

+ Γ119896

119901119894119862119894119895119901

+ Γ119894

119901119894119862119901119896119895

) (108)


= 119892119901119895

119892119902119896

119878119894

119901119902+

119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896


Γ119895

119901119894119862119894119901119896

= Γ119895

119894119901119862119894119901119896

=

1

2

(Γ119895

119894119901119862119894119901119896

+ Γ119895

119901119894119862119901119894119896

) =

1

2

119878119895

119894119901119862119901119896119894

= minus

1

2

119878119895

119901119894119862119896119901119894

(109)

similarly we obtain

Γ119896

119901119894119862119894119895119901

= Γ119896

119894119901119862119901119895119894

=

1

2

119862119895119901119894

(Γ119896

119894119901minus Γ119896

119901119894) =

1

2

119878119896

119894119901119862119895119901119894

=

1

2

119878119896

119901119902119862119895119902119901

(110)

Then we write

119862119894119896119895

119894= minus119862119894119895119896

119894

= minus119862119894119895119896

119894minus

1

2

119878119895

119901119902119862119896119901119902

+

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894+

1

2

119878119895

119901119902119862119896119901119902

minus

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

(111)



119901

119901119897=

(12)119892119894119901119897119892119894119901

= (1radic119892)(120597radic119892120597119909119897

) and Γ119901

119897119901= Γ119901

119901119897+ 119878119901

119897119901and

obtain

Γ119901

119901119897=

1

radicminus119892


120597119909119897

+ (ln120595)119897= (ln (120595radicminus119892))

119897 (112)

Then we obtain

119867119895119896

minus 119867119896119895

minus 119865119895119896

= minus119862119894119895119896


119894119862119894119896119895

(113)


120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


119894+ (ln (120595radicminus119892))

119894119862119894119896119895

)

120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


)119894

(114)


(120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

))119896

= 0 (115)







119894

119895119896 we



119894

119895119896and 119892

119894119896


119877119894119895= Γ119901

119894119901119895minus Γ119901

119894119895119901+ Γ119901

119902119895Γ119902

119894119901minus Γ119901

119902119901Γ119902

119894119895 (116)


119894119897119878119897

119896119895)119892119894119896


Γ119894

119895119896minus Γ119894


int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881 (117)




119877119894119896+ 119878119895

119894119897119878119897

119896119895= 0 (118)


119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901

minus 120575119895

119896(119892119894119901

119901minus

1

2

119892119894119901

119892119898119899119892119898119899

119901+ 119892119901119902

Γ119894

119901119902) minus 119892119894119895

Γ119901

119896119901

+ 2 (119892119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(119)



minus3119892119894119895

119895+

3

2

119892119894119895

119892119901119902119892119901119902

119895minus 3119892119901119902

Γ119894

119901119902+ 119892119894119901

119878119895

119901119895= 0 (120)

Therefore we have

119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901minus 119892119894119895

Γ119901

119896119901

minus

1

3

120575119895

119896119892119894119901

119878119902

119901119902+ 2 (119892

119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(121)


minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901+ 2 (119892

119901119895119878119901

119894119896+ 119892119894119901119878119901

119896119895) = 0

(122)



5 (119892119894119901119878119901

119896119895+ 119892119895119901119878119901

119894119896) +

1

3

(119892119895119896119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0 (123)

Further we obtained

minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901minus

2

15

(119892119896119895119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0

(124)


0 then from (123) we have 119878119894




119898+

119882119892) = 0 where 119882

119898and 119882




120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int (119877119894119896radicminus119892120575119892

119894119896

+ 119877119894119896119892119894119896

120575radicminus119892 + 119892119894119896

radicminus119892120575119877119894119896

+ 119878119895

119894119897119878119897

119896119895radicminus119892120575119892

119894119896

+ 119878119895

119894119897119878119897

119896119895119892119894119896

120575radicminus119892) 119889119881

119877119894119896119892119894119896


1

2

119877119901119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(125)

Similarly we obtain

119878119895

119894119897119878119897

119896119895119892119894119896


1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(126)



119894119896

(120575Γ119897

119896119894)119897minus 119892119894119896

(120575Γ119897

119896119897)119894= (119892119894119896

120575Γ119897

119896119894minus 119892119894119897

120575Γ119901

119897119901)119897



119894119896= 119892119894119896

120575(Γ119901

119902119901Γ119902

119896119894minus

Γ119901

119902119894Γ119902119897



119892119894119896

120575119877119894119896= (119878119901

119894119901119878119902

119902119896minus 119878119901

119894119902119878119902

119901119896minus 119892119896119901119892119902119905

119878119898

119894119905119878119901

119902119898) 120575119892119894119896

(127)

Thus we have

120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(128)


120575119882119892

= 1198701int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= 1198701int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(129)


1198883


120575119882119892= 1198702int119879119894119896radicminus119892120575119892

119894119896

119889119881 (130)


2usually


119892+120575119882119898=

0 we find relations

int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902


119894119896

119889119881 = 0

(131)



119877119894119896minus

1

2

119892119894119896119877 + 119878119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902+ 119878119895

119894119897119878119897

119896119895

minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896= 119870119879119894119896

(132)


2

7 Conclusions







References




























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119894119895


119894119895 then let 119872119901

119896119897equiv

(12)119892119901119894

(119892119896119898119878119898

119897119894+ 119892119897119898119878119898

119896119894) then119872119897

119896119897equiv (12)119878

119897

119896119897

We obtained

Γ119896

119894119895minus 119875119896

119894119895=

1

2

1

119899 + 1

(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897

119894119897) (67)



119894119895


119894119895=

(12)(1(119899 + 1))(120575119896

119894119878119897

119895119897+ 120575119896

119895119878119897



119896

119894119895119860119894

119889119909119895



119894119895







119894119895= 0 Assume that 119894 119872 sub 119875


119909rarr

119879119909119875|119872

rarr 119879perp



perp



119883119884 = 119879(nabla

1015840

119883119884) + perp (nabla

1015840

119883119884) Let it be

nabla119883119884 = 119879(nabla

1015840

119883119884) 120572(119883 119884) = perp (nabla




⟨1198771015840

(119883 119884)119885119882⟩ = ⟨119877 (119883 119884)119885119882⟩

+ ⟨120572 (119883 119885) 120572 (119884119882)⟩

minus ⟨120572 (119884 119885) 120572 (119883119882)⟩

(68)



(nabla10158401015840

119883120572) (119884 119885) = 119863

119883(120572 (119884 119885)) minus 120572 (nabla

119883119884119885)

minus 120572 (119884 nabla119883119885)

(69)

Codazzi equation is

perp ⟨1198771015840

(119883 119884)119885⟩ = (nabla10158401015840

119883120572) (119884 119885) minus (nabla

10158401015840

119884120572) (119883 119885) (70)





119909119894

= 119909119894

(1199101

119910119899minus1

) (71)


119886120572120573

= 119892119894119895

120597119909119894

120597119910120572

120597119909119895

120597119910120573

(72)


120572120573of



we assume that 119866120572120573120574



120572120573


119894119895and

119878119896

119894119895 we have

119866120572

120573120574

=

1

2

(119886120572120578

(119886120573120578120574

+ 119886120574120578120573

+ 119886120573120574120578

+ 119886120573120583119879120583

120574120578+ 119886120574120583119879120583

120573120578) + 119879120572

120574120573)

(73)


119866120572



1015840

by the formula

119866120572

120573120574= 1198661205721015840

12057310158401205741015840

120597119906120572

1205971199061205721015840

1205971199061205731015840

120597119906120573

1205971199061205741015840

120597119906120574+

120597119906120572

1205971199061205721015840

1205972

1199061205721015840

120597119906120573120597119906120574

(74)




119866120572

120573120574

120597119909119896

120597119906120572= Γ119896

119894119895

120597119909119894

120597119906120573

120597119909119895

120597119906120574+

1205972

119909119896

120597119906120573120597119906120574

(75)

We obtain

1

2

(119886120572120578

(119886120573120578120574

+ 119886120574120578120573

+ 119886120573120574120578

+ 119886120573120583119879120583

120574120578+ 119886120574120583119879120583

120573120578) + 119879120572

120574120573)

120597119909119896

120597119906120572

=

1

2

(119892119896119899

(119892119899119894119895

+ 119892119899119895119894

minus 119892119894119895119899

+ 119892119894119898119878119898

119895119899+ 119892119895119898119878119898

119894119899) + 119878119896

119894119895)

times

120597119909119894

120597119906120573

120597119909119895

120597119906120574+

1205972

119909119896

120597119906120573120597119906120574

(76)


119879120574

120572120573= 119886120574120578

119892119901119902119878119901

119894119895

120597119909119894

120597119910120572

120597119909119895

120597119910120573

120597119909119902

120597119910120578 (77)

using tensors 119886120572120573

and 119879120574












119860119894120572

119895120573120574= 119860119894120572

119895120573120574+ Γ119894

119901119896119860119901120572

119895120573

120597119909119896

120597119910120574minus Γ119901

119895119902119860119894120572

119901120573

120597119909119902

120597119910120574+ 119866120572

120578120574119860119894120578

119895120573

minus 119866120578

120573120574119860119894120572

119895120578

(78)


119906119894120572120573

minus 119906119894120573120572

= 119877119901

119896119897119894119906119901

120597119909119897

120597119910120572

120597119909119896

120597119910120573

+ 119878119902

119896119897119906119894119902

120597119909119897

120597119910120572

120597119909119896

120597119910120573

119906120574120572120573

minus 119906120574120573120572

= 119877120578

120573120572120574119906120578+ 119879120578

120573120572119906120574120578

(79)

where 119877120578



120573120574


120585119894

120572=

120597119909119894

120597119910120572 (80)



120585119894

1 120585

119894

119899minus1 ]119894 (81)

where 120585119894

1 120585

119894




120572

120585119894

120572120574= 120585119894

120572120574+ Γ119894

119901119902120585119901

120572

120597119909119902

120597119910120574minus 119866120578

120572120574120585119894

120578 (82)


120585119894

120572120574minus 120585119894

120574120572= 119878119894

119901119902120585119901

120572120585119902

120574+ 119879120578

120574120572120585119894

120578 (83)


0 = 119886120572120573120574

= (119892119894119895120585119894

120572120585119895

120573)

120574

= 119892119894119895120585119894

120572120574120585119895

120573+ 119892119894119895120585119894

120572120585119895

120573120574 (84)

we obtain

119892119894119895120585119894

120572120574120585119895

120573= 0 (85)


120585119894

120573120572= 120587120572120573]119894 (86)

Remark 11 120587120572120573



120574) the formula

119892119894119895]119894120574120585119894

120572= minus120587120574120572 (87)


]119894120574= minus119886120578120583

120587120583120574120585119894

120578 (88)



Further we obtain

120585119894

120573120594120582minus 120585119894

120573120582120594= minus119877119894

119896119897119901120585119896

120582120585119897

120594120585119901

120573+ 119877120590

120582120594120573120585119894

120590+ 119879120590

120582120594120585119894

120573120590

= (120587120594120573120582

minus 120587120582120573120594

) ]119894

minus (120587120594120573120587120578120582119886120578120590

minus 120587120582120573120587120578120594119886120578120590

) 120585119894

120590

(89)


120572 we have

119877120572120582120594120573

= 119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573120585119894

120572minus (120587120594120573120587120572120582

minus 120587120582120573120587120572120594) (90)


]119894120594120582

minus ]119894120582120594

= minus119877119894

119896119897119901120585119896

120582120585119897

120594]119901 + 119879

120590

120582120594]119894120590

= (120587120578120582120594

119886120578120590

minus 120587120578120594120582

119886120578120590

) 120585119894

120590

(91)


minus119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573]119894 + 119879

120590

120582120594120587120590120573

= 120587120594120573120582

minus 120587120582120573120594

(92)



minus119877119894119896119897119901

120585119896

120582120585119897

120594]119901120585119894120572+ 119879120590

120582120594120587120572120590

= 120587120572120582120594

minus 120587120572120594120582

(93)










119878119894

119895119896119901119902minus 119878119894

119895119896119902119901= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

+ 119878119905

119902119901119878119894

119895119896119905

(94)

or

119878119894

119895119896119901119902minus 119878119894

119895119896119902119901minus 119878119905

119902119901119878119894

119895119896119905

= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

(95)


119878119894

119895119896119901119894minus 119878119894

119895119896119894119901minus 119878119905

119894119901119878119894

119895119896119905

= (119878119894

119895119896119901minus 119878119894

119902119901119878119902

119895119896)

119894

minus 119878119894

119895119896119894119901minus 119878119894

119901119902119894119878119902

119895119896

(96)


(119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896)119894

minus 119892119896119901

119892119895119904

119878119894

119904119896119894119901minus 119892119896119901

119892119895119904

119878119894

119901119902119894119878119902

119904119896

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905

minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(97)


119867119895119894

= 119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896

119865119895119901

= 119892119896119901

119892119895119904

119878119894

119904119896119894

(98)


119867119895119894

119894minus 119865119895119894

119894minus 119892119896119901

119892119895119904

119878119902

119904119896119865119901119902

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(99)

where 119865119901119902

= 119878119894

119901119902119894


119894119896119897+ 119877119901

119896119897119894+ 119877119901

119897119894119896= 0 hence

119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896+ 119878119901

119897119902119878119902

119894119896+ 119878119901

119896119902119878119902

119897119894+ 119878119901

119894119902119878119902

119896119897= 0 (100)


119878119901

119894119896119901+ 119878119901

119896119901119894+ 119878119901

119901119894119896= 0 (101)




119894119895119878119902


expression

119878119901

119894119895119901= 120593119894119895minus 120593119895119894 (102)

Next if we put 119878119901119894119895119901


=



119894 System (99)

takes the form

119867119895119894

119894= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

119865119894119895

= 0

(103)


119862119894119895119896

= 119892119901119895

119892119902119896

119878119894

119901119902+ 119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896

119901119902 (104)


119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+ 119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904

minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

(105)



119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

=

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

(106)

hence

119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902) (107)


119862119894119896119895

119894= minus119862119894119895119896

119894= minus (119862

119894119895119896

119894+ Γ119895

119901119894119862119894119901119896

+ Γ119896

119901119894119862119894119895119901

+ Γ119894

119901119894119862119901119896119895

) (108)


= 119892119901119895

119892119902119896

119878119894

119901119902+

119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896


Γ119895

119901119894119862119894119901119896

= Γ119895

119894119901119862119894119901119896

=

1

2

(Γ119895

119894119901119862119894119901119896

+ Γ119895

119901119894119862119901119894119896

) =

1

2

119878119895

119894119901119862119901119896119894

= minus

1

2

119878119895

119901119894119862119896119901119894

(109)

similarly we obtain

Γ119896

119901119894119862119894119895119901

= Γ119896

119894119901119862119901119895119894

=

1

2

119862119895119901119894

(Γ119896

119894119901minus Γ119896

119901119894) =

1

2

119878119896

119894119901119862119895119901119894

=

1

2

119878119896

119901119902119862119895119902119901

(110)

Then we write

119862119894119896119895

119894= minus119862119894119895119896

119894

= minus119862119894119895119896

119894minus

1

2

119878119895

119901119902119862119896119901119902

+

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894+

1

2

119878119895

119901119902119862119896119901119902

minus

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

(111)



119901

119901119897=

(12)119892119894119901119897119892119894119901

= (1radic119892)(120597radic119892120597119909119897

) and Γ119901

119897119901= Γ119901

119901119897+ 119878119901

119897119901and

obtain

Γ119901

119901119897=

1

radicminus119892


120597119909119897

+ (ln120595)119897= (ln (120595radicminus119892))

119897 (112)

Then we obtain

119867119895119896

minus 119867119896119895

minus 119865119895119896

= minus119862119894119895119896


119894119862119894119896119895

(113)


120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


119894+ (ln (120595radicminus119892))

119894119862119894119896119895

)

120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


)119894

(114)


(120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

))119896

= 0 (115)







119894

119895119896 we



119894

119895119896and 119892

119894119896


119877119894119895= Γ119901

119894119901119895minus Γ119901

119894119895119901+ Γ119901

119902119895Γ119902

119894119901minus Γ119901

119902119901Γ119902

119894119895 (116)


119894119897119878119897

119896119895)119892119894119896


Γ119894

119895119896minus Γ119894


int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881 (117)




119877119894119896+ 119878119895

119894119897119878119897

119896119895= 0 (118)


119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901

minus 120575119895

119896(119892119894119901

119901minus

1

2

119892119894119901

119892119898119899119892119898119899

119901+ 119892119901119902

Γ119894

119901119902) minus 119892119894119895

Γ119901

119896119901

+ 2 (119892119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(119)



minus3119892119894119895

119895+

3

2

119892119894119895

119892119901119902119892119901119902

119895minus 3119892119901119902

Γ119894

119901119902+ 119892119894119901

119878119895

119901119895= 0 (120)

Therefore we have

119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901minus 119892119894119895

Γ119901

119896119901

minus

1

3

120575119895

119896119892119894119901

119878119902

119901119902+ 2 (119892

119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(121)


minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901+ 2 (119892

119901119895119878119901

119894119896+ 119892119894119901119878119901

119896119895) = 0

(122)



5 (119892119894119901119878119901

119896119895+ 119892119895119901119878119901

119894119896) +

1

3

(119892119895119896119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0 (123)

Further we obtained

minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901minus

2

15

(119892119896119895119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0

(124)


0 then from (123) we have 119878119894




119898+

119882119892) = 0 where 119882

119898and 119882




120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int (119877119894119896radicminus119892120575119892

119894119896

+ 119877119894119896119892119894119896

120575radicminus119892 + 119892119894119896

radicminus119892120575119877119894119896

+ 119878119895

119894119897119878119897

119896119895radicminus119892120575119892

119894119896

+ 119878119895

119894119897119878119897

119896119895119892119894119896

120575radicminus119892) 119889119881

119877119894119896119892119894119896


1

2

119877119901119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(125)

Similarly we obtain

119878119895

119894119897119878119897

119896119895119892119894119896


1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(126)



119894119896

(120575Γ119897

119896119894)119897minus 119892119894119896

(120575Γ119897

119896119897)119894= (119892119894119896

120575Γ119897

119896119894minus 119892119894119897

120575Γ119901

119897119901)119897



119894119896= 119892119894119896

120575(Γ119901

119902119901Γ119902

119896119894minus

Γ119901

119902119894Γ119902119897



119892119894119896

120575119877119894119896= (119878119901

119894119901119878119902

119902119896minus 119878119901

119894119902119878119902

119901119896minus 119892119896119901119892119902119905

119878119898

119894119905119878119901

119902119898) 120575119892119894119896

(127)

Thus we have

120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(128)


120575119882119892

= 1198701int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= 1198701int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(129)


1198883


120575119882119892= 1198702int119879119894119896radicminus119892120575119892

119894119896

119889119881 (130)


2usually


119892+120575119882119898=

0 we find relations

int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902


119894119896

119889119881 = 0

(131)



119877119894119896minus

1

2

119892119894119896119877 + 119878119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902+ 119878119895

119894119897119878119897

119896119895

minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896= 119870119879119894119896

(132)


2

7 Conclusions







References




























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




119866120572



1015840

by the formula

119866120572

120573120574= 1198661205721015840

12057310158401205741015840

120597119906120572

1205971199061205721015840

1205971199061205731015840

120597119906120573

1205971199061205741015840

120597119906120574+

120597119906120572

1205971199061205721015840

1205972

1199061205721015840

120597119906120573120597119906120574

(74)




119866120572

120573120574

120597119909119896

120597119906120572= Γ119896

119894119895

120597119909119894

120597119906120573

120597119909119895

120597119906120574+

1205972

119909119896

120597119906120573120597119906120574

(75)

We obtain

1

2

(119886120572120578

(119886120573120578120574

+ 119886120574120578120573

+ 119886120573120574120578

+ 119886120573120583119879120583

120574120578+ 119886120574120583119879120583

120573120578) + 119879120572

120574120573)

120597119909119896

120597119906120572

=

1

2

(119892119896119899

(119892119899119894119895

+ 119892119899119895119894

minus 119892119894119895119899

+ 119892119894119898119878119898

119895119899+ 119892119895119898119878119898

119894119899) + 119878119896

119894119895)

times

120597119909119894

120597119906120573

120597119909119895

120597119906120574+

1205972

119909119896

120597119906120573120597119906120574

(76)


119879120574

120572120573= 119886120574120578

119892119901119902119878119901

119894119895

120597119909119894

120597119910120572

120597119909119895

120597119910120573

120597119909119902

120597119910120578 (77)

using tensors 119886120572120573

and 119879120574












119860119894120572

119895120573120574= 119860119894120572

119895120573120574+ Γ119894

119901119896119860119901120572

119895120573

120597119909119896

120597119910120574minus Γ119901

119895119902119860119894120572

119901120573

120597119909119902

120597119910120574+ 119866120572

120578120574119860119894120578

119895120573

minus 119866120578

120573120574119860119894120572

119895120578

(78)


119906119894120572120573

minus 119906119894120573120572

= 119877119901

119896119897119894119906119901

120597119909119897

120597119910120572

120597119909119896

120597119910120573

+ 119878119902

119896119897119906119894119902

120597119909119897

120597119910120572

120597119909119896

120597119910120573

119906120574120572120573

minus 119906120574120573120572

= 119877120578

120573120572120574119906120578+ 119879120578

120573120572119906120574120578

(79)

where 119877120578



120573120574


120585119894

120572=

120597119909119894

120597119910120572 (80)



120585119894

1 120585

119894

119899minus1 ]119894 (81)

where 120585119894

1 120585

119894




120572

120585119894

120572120574= 120585119894

120572120574+ Γ119894

119901119902120585119901

120572

120597119909119902

120597119910120574minus 119866120578

120572120574120585119894

120578 (82)


120585119894

120572120574minus 120585119894

120574120572= 119878119894

119901119902120585119901

120572120585119902

120574+ 119879120578

120574120572120585119894

120578 (83)


0 = 119886120572120573120574

= (119892119894119895120585119894

120572120585119895

120573)

120574

= 119892119894119895120585119894

120572120574120585119895

120573+ 119892119894119895120585119894

120572120585119895

120573120574 (84)

we obtain

119892119894119895120585119894

120572120574120585119895

120573= 0 (85)


120585119894

120573120572= 120587120572120573]119894 (86)

Remark 11 120587120572120573



120574) the formula

119892119894119895]119894120574120585119894

120572= minus120587120574120572 (87)


]119894120574= minus119886120578120583

120587120583120574120585119894

120578 (88)



Further we obtain

120585119894

120573120594120582minus 120585119894

120573120582120594= minus119877119894

119896119897119901120585119896

120582120585119897

120594120585119901

120573+ 119877120590

120582120594120573120585119894

120590+ 119879120590

120582120594120585119894

120573120590

= (120587120594120573120582

minus 120587120582120573120594

) ]119894

minus (120587120594120573120587120578120582119886120578120590

minus 120587120582120573120587120578120594119886120578120590

) 120585119894

120590

(89)


120572 we have

119877120572120582120594120573

= 119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573120585119894

120572minus (120587120594120573120587120572120582

minus 120587120582120573120587120572120594) (90)


]119894120594120582

minus ]119894120582120594

= minus119877119894

119896119897119901120585119896

120582120585119897

120594]119901 + 119879

120590

120582120594]119894120590

= (120587120578120582120594

119886120578120590

minus 120587120578120594120582

119886120578120590

) 120585119894

120590

(91)


minus119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573]119894 + 119879

120590

120582120594120587120590120573

= 120587120594120573120582

minus 120587120582120573120594

(92)



minus119877119894119896119897119901

120585119896

120582120585119897

120594]119901120585119894120572+ 119879120590

120582120594120587120572120590

= 120587120572120582120594

minus 120587120572120594120582

(93)










119878119894

119895119896119901119902minus 119878119894

119895119896119902119901= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

+ 119878119905

119902119901119878119894

119895119896119905

(94)

or

119878119894

119895119896119901119902minus 119878119894

119895119896119902119901minus 119878119905

119902119901119878119894

119895119896119905

= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

(95)


119878119894

119895119896119901119894minus 119878119894

119895119896119894119901minus 119878119905

119894119901119878119894

119895119896119905

= (119878119894

119895119896119901minus 119878119894

119902119901119878119902

119895119896)

119894

minus 119878119894

119895119896119894119901minus 119878119894

119901119902119894119878119902

119895119896

(96)


(119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896)119894

minus 119892119896119901

119892119895119904

119878119894

119904119896119894119901minus 119892119896119901

119892119895119904

119878119894

119901119902119894119878119902

119904119896

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905

minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(97)


119867119895119894

= 119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896

119865119895119901

= 119892119896119901

119892119895119904

119878119894

119904119896119894

(98)


119867119895119894

119894minus 119865119895119894

119894minus 119892119896119901

119892119895119904

119878119902

119904119896119865119901119902

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(99)

where 119865119901119902

= 119878119894

119901119902119894


119894119896119897+ 119877119901

119896119897119894+ 119877119901

119897119894119896= 0 hence

119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896+ 119878119901

119897119902119878119902

119894119896+ 119878119901

119896119902119878119902

119897119894+ 119878119901

119894119902119878119902

119896119897= 0 (100)


119878119901

119894119896119901+ 119878119901

119896119901119894+ 119878119901

119901119894119896= 0 (101)




119894119895119878119902


expression

119878119901

119894119895119901= 120593119894119895minus 120593119895119894 (102)

Next if we put 119878119901119894119895119901


=



119894 System (99)

takes the form

119867119895119894

119894= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

119865119894119895

= 0

(103)


119862119894119895119896

= 119892119901119895

119892119902119896

119878119894

119901119902+ 119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896

119901119902 (104)


119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+ 119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904

minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

(105)



119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

=

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

(106)

hence

119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902) (107)


119862119894119896119895

119894= minus119862119894119895119896

119894= minus (119862

119894119895119896

119894+ Γ119895

119901119894119862119894119901119896

+ Γ119896

119901119894119862119894119895119901

+ Γ119894

119901119894119862119901119896119895

) (108)


= 119892119901119895

119892119902119896

119878119894

119901119902+

119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896


Γ119895

119901119894119862119894119901119896

= Γ119895

119894119901119862119894119901119896

=

1

2

(Γ119895

119894119901119862119894119901119896

+ Γ119895

119901119894119862119901119894119896

) =

1

2

119878119895

119894119901119862119901119896119894

= minus

1

2

119878119895

119901119894119862119896119901119894

(109)

similarly we obtain

Γ119896

119901119894119862119894119895119901

= Γ119896

119894119901119862119901119895119894

=

1

2

119862119895119901119894

(Γ119896

119894119901minus Γ119896

119901119894) =

1

2

119878119896

119894119901119862119895119901119894

=

1

2

119878119896

119901119902119862119895119902119901

(110)

Then we write

119862119894119896119895

119894= minus119862119894119895119896

119894

= minus119862119894119895119896

119894minus

1

2

119878119895

119901119902119862119896119901119902

+

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894+

1

2

119878119895

119901119902119862119896119901119902

minus

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

(111)



119901

119901119897=

(12)119892119894119901119897119892119894119901

= (1radic119892)(120597radic119892120597119909119897

) and Γ119901

119897119901= Γ119901

119901119897+ 119878119901

119897119901and

obtain

Γ119901

119901119897=

1

radicminus119892


120597119909119897

+ (ln120595)119897= (ln (120595radicminus119892))

119897 (112)

Then we obtain

119867119895119896

minus 119867119896119895

minus 119865119895119896

= minus119862119894119895119896


119894119862119894119896119895

(113)


120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


119894+ (ln (120595radicminus119892))

119894119862119894119896119895

)

120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


)119894

(114)


(120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

))119896

= 0 (115)







119894

119895119896 we



119894

119895119896and 119892

119894119896


119877119894119895= Γ119901

119894119901119895minus Γ119901

119894119895119901+ Γ119901

119902119895Γ119902

119894119901minus Γ119901

119902119901Γ119902

119894119895 (116)


119894119897119878119897

119896119895)119892119894119896


Γ119894

119895119896minus Γ119894


int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881 (117)




119877119894119896+ 119878119895

119894119897119878119897

119896119895= 0 (118)


119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901

minus 120575119895

119896(119892119894119901

119901minus

1

2

119892119894119901

119892119898119899119892119898119899

119901+ 119892119901119902

Γ119894

119901119902) minus 119892119894119895

Γ119901

119896119901

+ 2 (119892119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(119)



minus3119892119894119895

119895+

3

2

119892119894119895

119892119901119902119892119901119902

119895minus 3119892119901119902

Γ119894

119901119902+ 119892119894119901

119878119895

119901119895= 0 (120)

Therefore we have

119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901minus 119892119894119895

Γ119901

119896119901

minus

1

3

120575119895

119896119892119894119901

119878119902

119901119902+ 2 (119892

119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(121)


minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901+ 2 (119892

119901119895119878119901

119894119896+ 119892119894119901119878119901

119896119895) = 0

(122)



5 (119892119894119901119878119901

119896119895+ 119892119895119901119878119901

119894119896) +

1

3

(119892119895119896119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0 (123)

Further we obtained

minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901minus

2

15

(119892119896119895119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0

(124)


0 then from (123) we have 119878119894




119898+

119882119892) = 0 where 119882

119898and 119882




120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int (119877119894119896radicminus119892120575119892

119894119896

+ 119877119894119896119892119894119896

120575radicminus119892 + 119892119894119896

radicminus119892120575119877119894119896

+ 119878119895

119894119897119878119897

119896119895radicminus119892120575119892

119894119896

+ 119878119895

119894119897119878119897

119896119895119892119894119896

120575radicminus119892) 119889119881

119877119894119896119892119894119896


1

2

119877119901119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(125)

Similarly we obtain

119878119895

119894119897119878119897

119896119895119892119894119896


1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(126)



119894119896

(120575Γ119897

119896119894)119897minus 119892119894119896

(120575Γ119897

119896119897)119894= (119892119894119896

120575Γ119897

119896119894minus 119892119894119897

120575Γ119901

119897119901)119897



119894119896= 119892119894119896

120575(Γ119901

119902119901Γ119902

119896119894minus

Γ119901

119902119894Γ119902119897



119892119894119896

120575119877119894119896= (119878119901

119894119901119878119902

119902119896minus 119878119901

119894119902119878119902

119901119896minus 119892119896119901119892119902119905

119878119898

119894119905119878119901

119902119898) 120575119892119894119896

(127)

Thus we have

120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(128)


120575119882119892

= 1198701int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= 1198701int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(129)


1198883


120575119882119892= 1198702int119879119894119896radicminus119892120575119892

119894119896

119889119881 (130)


2usually


119892+120575119882119898=

0 we find relations

int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902


119894119896

119889119881 = 0

(131)



119877119894119896minus

1

2

119892119894119896119877 + 119878119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902+ 119878119895

119894119897119878119897

119896119895

minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896= 119870119879119894119896

(132)


2

7 Conclusions







References




























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Further we obtain

120585119894

120573120594120582minus 120585119894

120573120582120594= minus119877119894

119896119897119901120585119896

120582120585119897

120594120585119901

120573+ 119877120590

120582120594120573120585119894

120590+ 119879120590

120582120594120585119894

120573120590

= (120587120594120573120582

minus 120587120582120573120594

) ]119894

minus (120587120594120573120587120578120582119886120578120590

minus 120587120582120573120587120578120594119886120578120590

) 120585119894

120590

(89)


120572 we have

119877120572120582120594120573

= 119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573120585119894

120572minus (120587120594120573120587120572120582

minus 120587120582120573120587120572120594) (90)


]119894120594120582

minus ]119894120582120594

= minus119877119894

119896119897119901120585119896

120582120585119897

120594]119901 + 119879

120590

120582120594]119894120590

= (120587120578120582120594

119886120578120590

minus 120587120578120594120582

119886120578120590

) 120585119894

120590

(91)


minus119877119894119896119897119901

120585119896

120582120585119897

120594120585119901

120573]119894 + 119879

120590

120582120594120587120590120573

= 120587120594120573120582

minus 120587120582120573120594

(92)



minus119877119894119896119897119901

120585119896

120582120585119897

120594]119901120585119894120572+ 119879120590

120582120594120587120572120590

= 120587120572120582120594

minus 120587120572120594120582

(93)










119878119894

119895119896119901119902minus 119878119894

119895119896119902119901= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

+ 119878119905

119902119901119878119894

119895119896119905

(94)

or

119878119894

119895119896119901119902minus 119878119894

119895119896119902119901minus 119878119905

119902119901119878119894

119895119896119905

= 119877119905

119902119901119895119878119894

119905119896+ 119877119905

119902119901119896119878119894

119895119905minus 119877119894

119902119901119905119878119905

119895119896

(95)


119878119894

119895119896119901119894minus 119878119894

119895119896119894119901minus 119878119905

119894119901119878119894

119895119896119905

= (119878119894

119895119896119901minus 119878119894

119902119901119878119902

119895119896)

119894

minus 119878119894

119895119896119894119901minus 119878119894

119901119902119894119878119902

119895119896

(96)


(119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896)119894

minus 119892119896119901

119892119895119904

119878119894

119904119896119894119901minus 119892119896119901

119892119895119904

119878119894

119901119902119894119878119902

119904119896

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905

minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(97)


119867119895119894

= 119892119896119901

119892119895119904

119878119894

119904119896119901minus 119892119896119901

119892119895119904

119878119894

119902119901119878119902

119904119896

119865119895119901

= 119892119896119901

119892119895119904

119878119894

119904119896119894

(98)


119867119895119894

119894minus 119865119895119894

119894minus 119892119896119901

119892119895119904

119878119902

119904119896119865119901119902

= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

(99)

where 119865119901119902

= 119878119894

119901119902119894


119894119896119897+ 119877119901

119896119897119894+ 119877119901

119897119894119896= 0 hence

119878119901

119894119896119897+ 119878119901

119896119897119894+ 119878119901

119897119894119896+ 119878119901

119897119902119878119902

119894119896+ 119878119901

119896119902119878119902

119897119894+ 119878119901

119894119902119878119902

119896119897= 0 (100)


119878119901

119894119896119901+ 119878119901

119896119901119894+ 119878119901

119901119894119896= 0 (101)




119894119895119878119902


expression

119878119901

119894119895119901= 120593119894119895minus 120593119895119894 (102)

Next if we put 119878119901119894119895119901


=



119894 System (99)

takes the form

119867119895119894

119894= 119892119896119901

119892119895119904

119877119905

119894119901119904119878119894

119905119896+ 119892119896119901

119892119895119904

119877119905

119894119901119896119878119894

119904119905minus 119892119896119901

119892119895119904

119877119894

119894119901119905119878119905

119904119896

119865119894119895

= 0

(103)


119862119894119895119896

= 119892119901119895

119892119902119896

119878119894

119901119902+ 119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896

119901119902 (104)


119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+ 119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904

minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

(105)



119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

=

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

(106)

hence

119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902) (107)


119862119894119896119895

119894= minus119862119894119895119896

119894= minus (119862

119894119895119896

119894+ Γ119895

119901119894119862119894119901119896

+ Γ119896

119901119894119862119894119895119901

+ Γ119894

119901119894119862119901119896119895

) (108)


= 119892119901119895

119892119902119896

119878119894

119901119902+

119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896


Γ119895

119901119894119862119894119901119896

= Γ119895

119894119901119862119894119901119896

=

1

2

(Γ119895

119894119901119862119894119901119896

+ Γ119895

119901119894119862119901119894119896

) =

1

2

119878119895

119894119901119862119901119896119894

= minus

1

2

119878119895

119901119894119862119896119901119894

(109)

similarly we obtain

Γ119896

119901119894119862119894119895119901

= Γ119896

119894119901119862119901119895119894

=

1

2

119862119895119901119894

(Γ119896

119894119901minus Γ119896

119901119894) =

1

2

119878119896

119894119901119862119895119901119894

=

1

2

119878119896

119901119902119862119895119902119901

(110)

Then we write

119862119894119896119895

119894= minus119862119894119895119896

119894

= minus119862119894119895119896

119894minus

1

2

119878119895

119901119902119862119896119901119902

+

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894+

1

2

119878119895

119901119902119862119896119901119902

minus

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

(111)



119901

119901119897=

(12)119892119894119901119897119892119894119901

= (1radic119892)(120597radic119892120597119909119897

) and Γ119901

119897119901= Γ119901

119901119897+ 119878119901

119897119901and

obtain

Γ119901

119901119897=

1

radicminus119892


120597119909119897

+ (ln120595)119897= (ln (120595radicminus119892))

119897 (112)

Then we obtain

119867119895119896

minus 119867119896119895

minus 119865119895119896

= minus119862119894119895119896


119894119862119894119896119895

(113)


120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


119894+ (ln (120595radicminus119892))

119894119862119894119896119895

)

120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


)119894

(114)


(120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

))119896

= 0 (115)







119894

119895119896 we



119894

119895119896and 119892

119894119896


119877119894119895= Γ119901

119894119901119895minus Γ119901

119894119895119901+ Γ119901

119902119895Γ119902

119894119901minus Γ119901

119902119901Γ119902

119894119895 (116)


119894119897119878119897

119896119895)119892119894119896


Γ119894

119895119896minus Γ119894


int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881 (117)




119877119894119896+ 119878119895

119894119897119878119897

119896119895= 0 (118)


119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901

minus 120575119895

119896(119892119894119901

119901minus

1

2

119892119894119901

119892119898119899119892119898119899

119901+ 119892119901119902

Γ119894

119901119902) minus 119892119894119895

Γ119901

119896119901

+ 2 (119892119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(119)



minus3119892119894119895

119895+

3

2

119892119894119895

119892119901119902119892119901119902

119895minus 3119892119901119902

Γ119894

119901119902+ 119892119894119901

119878119895

119901119895= 0 (120)

Therefore we have

119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901minus 119892119894119895

Γ119901

119896119901

minus

1

3

120575119895

119896119892119894119901

119878119902

119901119902+ 2 (119892

119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(121)


minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901+ 2 (119892

119901119895119878119901

119894119896+ 119892119894119901119878119901

119896119895) = 0

(122)



5 (119892119894119901119878119901

119896119895+ 119892119895119901119878119901

119894119896) +

1

3

(119892119895119896119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0 (123)

Further we obtained

minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901minus

2

15

(119892119896119895119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0

(124)


0 then from (123) we have 119878119894




119898+

119882119892) = 0 where 119882

119898and 119882




120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int (119877119894119896radicminus119892120575119892

119894119896

+ 119877119894119896119892119894119896

120575radicminus119892 + 119892119894119896

radicminus119892120575119877119894119896

+ 119878119895

119894119897119878119897

119896119895radicminus119892120575119892

119894119896

+ 119878119895

119894119897119878119897

119896119895119892119894119896

120575radicminus119892) 119889119881

119877119894119896119892119894119896


1

2

119877119901119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(125)

Similarly we obtain

119878119895

119894119897119878119897

119896119895119892119894119896


1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(126)



119894119896

(120575Γ119897

119896119894)119897minus 119892119894119896

(120575Γ119897

119896119897)119894= (119892119894119896

120575Γ119897

119896119894minus 119892119894119897

120575Γ119901

119897119901)119897



119894119896= 119892119894119896

120575(Γ119901

119902119901Γ119902

119896119894minus

Γ119901

119902119894Γ119902119897



119892119894119896

120575119877119894119896= (119878119901

119894119901119878119902

119902119896minus 119878119901

119894119902119878119902

119901119896minus 119892119896119901119892119902119905

119878119898

119894119905119878119901

119902119898) 120575119892119894119896

(127)

Thus we have

120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(128)


120575119882119892

= 1198701int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= 1198701int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(129)


1198883


120575119882119892= 1198702int119879119894119896radicminus119892120575119892

119894119896

119889119881 (130)


2usually


119892+120575119882119898=

0 we find relations

int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902


119894119896

119889119881 = 0

(131)



119877119894119896minus

1

2

119892119894119896119877 + 119878119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902+ 119878119895

119894119897119878119897

119896119895

minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896= 119870119879119894119896

(132)


2

7 Conclusions







References




























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119892119896119901

119892119902119904

119878119905

119901119902119878119895

119905119904minus 119892119895119901

119892119902119904

119878119905

119901119902119878119896

119905119904

=

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

(106)

hence

119867119895119896

minus 119867119896119895

= 119862119894119896119895

119894+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902) (107)


119862119894119896119895

119894= minus119862119894119895119896

119894= minus (119862

119894119895119896

119894+ Γ119895

119901119894119862119894119901119896

+ Γ119896

119901119894119862119894119895119901

+ Γ119894

119901119894119862119901119896119895

) (108)


= 119892119901119895

119892119902119896

119878119894

119901119902+

119892119901119896

119892119902119894

119878119895

119901119902+ 119892119901119894

119892119902119895

119878119896


Γ119895

119901119894119862119894119901119896

= Γ119895

119894119901119862119894119901119896

=

1

2

(Γ119895

119894119901119862119894119901119896

+ Γ119895

119901119894119862119901119894119896

) =

1

2

119878119895

119894119901119862119901119896119894

= minus

1

2

119878119895

119901119894119862119896119901119894

(109)

similarly we obtain

Γ119896

119901119894119862119894119895119901

= Γ119896

119894119901119862119901119895119894

=

1

2

119862119895119901119894

(Γ119896

119894119901minus Γ119896

119901119894) =

1

2

119878119896

119894119901119862119895119901119894

=

1

2

119878119896

119901119902119862119895119902119901

(110)

Then we write

119862119894119896119895

119894= minus119862119894119895119896

119894

= minus119862119894119895119896

119894minus

1

2

119878119895

119901119902119862119896119901119902

+

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894+

1

2

119878119895

119901119902119862119896119901119902

minus

1

2

119878119896

119901119902119862119895119901119902

minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

+

1

2

(119862119895119901119902

119878119896

119901119902minus 119862119896119901119902

119878119895

119901119902)

119867119895119896

minus 119867119896119895

= minus119862119894119895119896

119894minus Γ119902

119901119902119862119901119896119895

+ 119865119895119896

(111)



119901

119901119897=

(12)119892119894119901119897119892119894119901

= (1radic119892)(120597radic119892120597119909119897

) and Γ119901

119897119901= Γ119901

119901119897+ 119878119901

119897119901and

obtain

Γ119901

119901119897=

1

radicminus119892


120597119909119897

+ (ln120595)119897= (ln (120595radicminus119892))

119897 (112)

Then we obtain

119867119895119896

minus 119867119896119895

minus 119865119895119896

= minus119862119894119895119896


119894119862119894119896119895

(113)


120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


119894+ (ln (120595radicminus119892))

119894119862119894119896119895

)

120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

)


)119894

(114)


(120595radicminus119892 (119867119895119896

minus 119867119896119895

minus 119865119895119896

))119896

= 0 (115)







119894

119895119896 we



119894

119895119896and 119892

119894119896


119877119894119895= Γ119901

119894119901119895minus Γ119901

119894119895119901+ Γ119901

119902119895Γ119902

119894119901minus Γ119901

119902119901Γ119902

119894119895 (116)


119894119897119878119897

119896119895)119892119894119896


Γ119894

119895119896minus Γ119894


int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881 (117)




119877119894119896+ 119878119895

119894119897119878119897

119896119895= 0 (118)


119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901

minus 120575119895

119896(119892119894119901

119901minus

1

2

119892119894119901

119892119898119899119892119898119899

119901+ 119892119901119902

Γ119894

119901119902) minus 119892119894119895

Γ119901

119896119901

+ 2 (119892119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(119)



minus3119892119894119895

119895+

3

2

119892119894119895

119892119901119902119892119901119902

119895minus 3119892119901119902

Γ119894

119901119902+ 119892119894119901

119878119895

119901119895= 0 (120)

Therefore we have

119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901minus 119892119894119895

Γ119901

119896119901

minus

1

3

120575119895

119896119892119894119901

119878119902

119901119902+ 2 (119892

119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(121)


minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901+ 2 (119892

119901119895119878119901

119894119896+ 119892119894119901119878119901

119896119895) = 0

(122)



5 (119892119894119901119878119901

119896119895+ 119892119895119901119878119901

119894119896) +

1

3

(119892119895119896119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0 (123)

Further we obtained

minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901minus

2

15

(119892119896119895119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0

(124)


0 then from (123) we have 119878119894




119898+

119882119892) = 0 where 119882

119898and 119882




120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int (119877119894119896radicminus119892120575119892

119894119896

+ 119877119894119896119892119894119896

120575radicminus119892 + 119892119894119896

radicminus119892120575119877119894119896

+ 119878119895

119894119897119878119897

119896119895radicminus119892120575119892

119894119896

+ 119878119895

119894119897119878119897

119896119895119892119894119896

120575radicminus119892) 119889119881

119877119894119896119892119894119896


1

2

119877119901119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(125)

Similarly we obtain

119878119895

119894119897119878119897

119896119895119892119894119896


1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(126)



119894119896

(120575Γ119897

119896119894)119897minus 119892119894119896

(120575Γ119897

119896119897)119894= (119892119894119896

120575Γ119897

119896119894minus 119892119894119897

120575Γ119901

119897119901)119897



119894119896= 119892119894119896

120575(Γ119901

119902119901Γ119902

119896119894minus

Γ119901

119902119894Γ119902119897



119892119894119896

120575119877119894119896= (119878119901

119894119901119878119902

119902119896minus 119878119901

119894119902119878119902

119901119896minus 119892119896119901119892119902119905

119878119898

119894119905119878119901

119902119898) 120575119892119894119896

(127)

Thus we have

120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(128)


120575119882119892

= 1198701int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= 1198701int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(129)


1198883


120575119882119892= 1198702int119879119894119896radicminus119892120575119892

119894119896

119889119881 (130)


2usually


119892+120575119882119898=

0 we find relations

int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902


119894119896

119889119881 = 0

(131)



119877119894119896minus

1

2

119892119894119896119877 + 119878119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902+ 119878119895

119894119897119878119897

119896119895

minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896= 119870119879119894119896

(132)


2

7 Conclusions







References




























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





minus3119892119894119895

119895+

3

2

119892119894119895

119892119901119902119892119901119902

119895minus 3119892119901119902

Γ119894

119901119902+ 119892119894119901

119878119895

119901119895= 0 (120)

Therefore we have

119892119894119895

119896minus

1

2

119892119894119895

119892119901119902119892119901119902

119896+ 119892119901119895

Γ119894

119901119896+ 119892119894119901

Γ119895

119896119901minus 119892119894119895

Γ119901

119896119901

minus

1

3

120575119895

119896119892119894119901

119878119902

119901119902+ 2 (119892

119894119901

119878119895

119901119896+ 119892119901119895

119878119894

119896119901) = 0

(121)


minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901+ 2 (119892

119901119895119878119901

119894119896+ 119892119894119901119878119901

119896119895) = 0

(122)



5 (119892119894119901119878119901

119896119895+ 119892119895119901119878119901

119894119896) +

1

3

(119892119895119896119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0 (123)

Further we obtained

minus 119892119894119895119896

minus

1

2

119892119894119895119892119901119902

119892119901119902119896

+ 119892119894119901Γ119901

119896119895+ 119892119895119901Γ119901

119894119896minus 119892119894119895Γ119901

119896119901

minus

1

3

119892119895119896119878119901

119894119901minus

2

15

(119892119896119895119878119901

119894119901+ 119892119894119896119878119901

119895119901) = 0

(124)


0 then from (123) we have 119878119894




119898+

119882119892) = 0 where 119882

119898and 119882




120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int (119877119894119896radicminus119892120575119892

119894119896

+ 119877119894119896119892119894119896

120575radicminus119892 + 119892119894119896

radicminus119892120575119877119894119896

+ 119878119895

119894119897119878119897

119896119895radicminus119892120575119892

119894119896

+ 119878119895

119894119897119878119897

119896119895119892119894119896

120575radicminus119892) 119889119881

119877119894119896119892119894119896


1

2

119877119901119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(125)

Similarly we obtain

119878119895

119894119897119878119897

119896119895119892119894119896


1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896radicminus119892120575119892

119894119896

(126)



119894119896

(120575Γ119897

119896119894)119897minus 119892119894119896

(120575Γ119897

119896119897)119894= (119892119894119896

120575Γ119897

119896119894minus 119892119894119897

120575Γ119901

119897119901)119897



119894119896= 119892119894119896

120575(Γ119901

119902119901Γ119902

119896119894minus

Γ119901

119902119894Γ119902119897



119892119894119896

120575119877119894119896= (119878119901

119894119901119878119902

119902119896minus 119878119901

119894119902119878119902

119901119896minus 119892119896119901119892119902119905

119878119898

119894119905119878119901

119902119898) 120575119892119894119896

(127)

Thus we have

120575int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(128)


120575119882119892

= 1198701int (119877119894119896+ 119878119895

119894119897119878119897

119896119895) 119892119894119896

radicminus119892119889119881

= 1198701int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896)radicminus119892120575119892

119894119896

119889119881

(129)


1198883


120575119882119892= 1198702int119879119894119896radicminus119892120575119892

119894119896

119889119881 (130)


2usually


119892+120575119882119898=

0 we find relations

int(119877119894119896minus

1

2

119892119894119896119877 + (119878

119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902)

+ 119878119895

119894119897119878119897

119896119895minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902


119894119896

119889119881 = 0

(131)



119877119894119896minus

1

2

119892119894119896119877 + 119878119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902+ 119878119895

119894119897119878119897

119896119895

minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896= 119870119879119894119896

(132)


2

7 Conclusions







References




























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119877119894119896minus

1

2

119892119894119896119877 + 119878119901

119894119901119878119902

119902119896+ 119878119902

119894119901119878119901

119896119902+ 119892119896119901119892119902119905

119878119898

119894119905119878119901

119898119902+ 119878119895

119894119897119878119897

119896119895

minus

1

2

119878119895

119901119897119878119897

119896119902119892119901119902

119892119894119896= 119870119879119894119896

(132)


2

7 Conclusions







References




























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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