transformations of manifolds and application to differential equations

~ Pitman Monographs and I~ Surveys in Pure and Applied Mathematics 93
Transformations of manifolds and applications to differential equations
Keti Tenenblat
~LONGMAN
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0 Addison Wesley Longman Limited 1998
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First published 1998
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Contents
PREFACE ..•...............•.................•........................... vii
CHAPTER I - TRANSFORMATIONS OF SURFACES AND APPLICATIONS •...... 1 §1. The structure equations. . ........................................... 2 §2. Differential equations associated to linear Weingarten surfaces ......... 4 §3. Geodesic congruences and parallel surfaces .......................... 22 §4. Pseudo-spherical geodesic congruences ............................... 30 §5. Backlund transformation for the sine-Gordon and the
elliptic sinh-Gordon equations. Superposition formula ............... 43 §6. The Laplace Transformation for second-order hyperbolic
equations and its geometric interpretation .......................... 50 §7. Differential equations which describe pseudo-spherical surfaces ....... 57
CHAPTER II - SUBMANIFOLDS OF CONSTANT SECTIONAL CURVATURE ... 73 §1. The structure equations in a pseudo-Riemannian space form. . ....... 73 §2. Submanifolds of constant sectional curvature.
The Generating equation ........................................... 76
AND APPLICATIONS •••.•......•.................•.••..•.....•..... 84 §1. Pseudo-spherical geodesic congruences. A generalization
of Backlund's Theorem ............................................ 85 §2. Permutability Theorem ............................................. 97 §3. Backlund transformation and superposition formula for the Generalized
Wave equation and the Generalized sine-Gordon equation ......... 100 §4. Linearization of the Backlund transformation ....................... 104 §5. The inverse scattering method for the Generalized Wave equation ... 107 §6. The inverse scattering method for the Generalized
sine-Gordon equation ............................................. 112 §7. The Backlund transformation in terms of scattering data.
Soliton solutions ................................................. 115
CHAPTER IV - THE GENERATING EQUATION .•....•...........•....•.. 119 §1. The Generating equation ........................................... 119 §2. Backlund transformation for the Generating equation
and its linearization ............. · · · · · · · · · · · · · · .................... 120 ~3. Superposition formula ................. · .. · ........................ 127
CHAPTER V - THE GENERATING INTRINSIC EQUATION ............... 131 §1. The Generating Intrinsic equation. Subma.nifolds
of constant curvature characterized by the metric. . ............... 132 §2. Backlund transformation for the Generating Intrinsic equation.
Symmetry group. . ............................................... 137 §3. Hyperbolic toroidal submanifolds of Euclidean space ................ 143 §4. Flat toroidal subma.nifolds of the unit sphere ....................... 151 §5. Geometric properties of submanifolds associated
to special solutions ............................................... 157
CHAPTER VI - LAPLACE TRANSFORMATION IN HIGHER DIMENSIONS .... 162 §1. Laplace transformations of Cartan manifolds ....................... 163 §2. The higher-dimensional Laplace invariants for systems
of second order PDEs ............................................. 172 §3. The generalized method of Laplace for systems of second
order PDEs ...................................................... 180 §4. Applications of the Laplace transformation to hydrodynamic
systems rich in conservation laws ................................. 188
REFERENCES ............•.................................•........... 201
INDEX ................................................................. 207
Preface
The interaction between differential geometry and partial differential equations has been studied since the last century and it can be found for example in the classical works of Lie, Darboux, Goursat, Bianchi, Backlund, E. Cartan. This relationship is based on the fact that most of the local properties of manifolds a.re expressed in terms of partial differential equations. Therefore, it is important to study transformations of manifolds which preserve such a geometric property, since the analytic interpretation of these transformations will provide mappings between the corresponding differential equations.
This correspondence between certain classes of manifolds and the associated differential equations can be useful in two ways. From our knowledge about the geometry of the manifolds we can obtain solutions to the equations. Conversely, we can obtain geometric properties of the manifolds or even prove the non existence of certain geometric structures on manifolds from our knowledge on the solutions of the corresponding differential equations.
One of the best known examples is the correspondence between surfaces of constant negative Gaussian curvature and solutions of the sine-Gordon equation. Backlund's classical transformation which takes a surface of constant negative cur vature into another such surface, when formulated analytically, defines a mapping taking solutions of the sine-Gordon equation into other solutions of the same equa tion. Backlund transformations have become a very important tool in the theory of soliton solutions of completely integrable equations. Bianchi's Permutability theo rem when interpreted analytically, defines a superposition formula which provides more solutions for the sine-Gordon equation algebraically.
On the other hand Hilbert's theorem, for example, which proves the nonexis tence of complete hyperbolic surfaces in Ef3, is a. consequence of the fact that the sine-Gordon equation does not have a solution defined on R2 whose image lies in the open interval (0, 1r). One can also obtain geometric properties of the surfaces associated to special solutions of the sine-Gordon equation.
This kind of interaction between differential geometry and differential equa tions will be the general theme of this book. Most of the differential equations we will be dealing with are highly nonlinear and hence difficult to solve. Therefore, it is important to mention the role played by differential geometry in the study of integrable differential equations not only in the classical examples but also in more recent results exhibiting integrablt: :systems with c:u1 arbitrary number of indepen dent variables. The classical Backlund transformation has regained attention in the last two decades due to the fact that it provides multi-soliton solutions for
V111
the sine-Gordon equation. The concept of soliton solution for nonlinear evolution equation has been of great interest to physicists and mathematicians since it. has the property of preserving its shape after a collision with other such solutions.
A generalizaton of the classical results for hyperbolic surfaces led first t.o a gene ralized sine-Gordon equation and a generalized wave equation and their Backlund transformations and superposition formulae. Later, similar results were obtained for the generalized elliptic sinh-Gordon equation and a. generalized Laplace equa tion. More recently, these multidimensional systems of differential equations were shown to be particular cases of a class of equations called Generating equation. So lutions of this equation correspond ton-dimensional submanifolds Af" of constant. sectional nuvature of semi-Riemannian space forms of dimension 2n- 1.
These and other results show t.he importance of the interaction between diffe rential geometry and differentia.! equations.
Our aim in this book is to present some of the classical theory on transforma tions of surfaces ar.d its more recent generalizations to higher dimensional mani folds with its applications to systems of partial differential equations. Although the classical theory may be found in the literature, its presentation aims to be instructive and motivating for the treatment of the theory in higher dimensions.
We now describe the contents of the book systematically. In Chapter I, we concentrate on the study of the two-dimensional case. We begin by characteri zing the differential equations which correspond to linear Weingarten surfaces in semi-Riemannian three-dimensional space forms. We show that such a surface, when it is parametrized by lines of curvatures, is locally determined by its metric which corresponds to a solution of one of the following differential equations: th.e elliptic or hyperbolic sine-Gordon, sinh-Gordon, cosh-Gordon or Liouville equa tion, the wave equation, or the Laplace equation. The ellipticity or hyperbolicity of the differential equations is determined by the index of the metric on the sur face. In particular. we associate surfaces of constant curvature in Riemannian space forms with solutions of the sine-Gordon equation, the elliptic sinh-Gordon, the wave and the Laplace equations. By considering geodesic congruences be tween parallel surfaces, we obtain Bonnet's theorem in Riemannian space forms and we consider parallel linear Weingarten surfaces. Pseudo-spherical geodesic
congruences on space forms of curvature K lead to a transformation between sur faces of the same constant Gaussian curvature /{ < N. The composition of such transformations are considered in Bianchi's Permutability Theorem. The a.na.:ytic interpretation of these transformations provides a. Backlund transformation and a superposition formula. for the sine-Gordon equation. Similar results are given for the elliptic sinh-Gordon equation by using analytic methods. Another interesting transformation we consider in this chapter is the Laplace transformation with its applications to second order hyperbolic eyu(l.tions c:tnd the meLhuu of sulving such equations by using their Laplace invariants. In the last section of this chapter, we introduce the notion of differential equations which describe pseudo-sphe:-ical
Preface IX
surfaces and we give a number of examples, most of them of physical interest. The geometry of such surfaces is used to obtain Backlund transformations and conservation laws for the differenticJ equa.tions.
Chapter II begins with a summary of the theory of moving frames for sub manifolds of semi-Riemannian space forms. We then consider n-dimensional sub manifolds Mn(l\) of coustant curvature A" isometrically immersed in a (2n- 1 )-
dimensional semi-Riemannian space form il.f of constant curvature K, such that
K -::f I<. Assuming the ~ormal bundle is flat and the principal normal curvatures
are different from A'- I\, we characterize such immersions in terms of a system of partial differential equa1.ions which are satisfied by an O(n - q, q) matrix valued fuction. This system of equations is called the Generating equation. This denomi nation is due to the fact that by choosing q and K it generates distinct differential equations.
In Chapter III, we restrict ourselves to those submanifolds !vr( K) of M 2n-t(I{)
for which K < /{ and its associated systems of equations for an 0( n) valued func tion which are called the Generalized sine-Gordor: equation (GSGE) if /{ :/: 0 and the Generalized wave equation (G\VE) if /{ = 0. We introduce the no-
tion of a pseudo-spherical geodesic congruence in M and derive a generalization of Backlund's theorem, which provides a transformation between submanifolds
Mn C M with the same constant sectional curvature. The composition of these transformations gives the Permutability theorem. The Backlund transformation and the superposition formula for the GSGE and the GWE are obtained from the analytic interpretation of the geometric results on pseudo-spherical geodesic con gruences and they provide soliton solutions for these equations. Co~sidering the linearization of the Backlund transformation we solve an initial-boundary value problem for both equations by using the inverse scattering method.
Chu.ptcr IV returns to the Generating equation. In contrast with the geometric point of view of the previous chapter, we extend the Backlund transformation, its linearization and the superposition formula to the Generating equation, by using analytic methods.
Motivated by the geometrically intrinsic aspect of the differei:tial equations which describe pseudo-spherical surfaces considered in Chapter I, we introduce the Generating Intrinsic equation in Chapter V. This is a subset of the Genera ting equation which is given by the Gauss equatio::t of a submanifold associated to a solution of the Generating equation. We show that solutions of the Gene rating Intrinsic equation are in correspondence with the solutions of the Genera ting equation. Therefore, we conclude that submanifolds of constant curvature M"(K) c M 2n-1(K) associated to solutions of the Generating equations are de termined up to a rigid moLiuu of M, by their metric which are solutions of the Generating Intrinsic equation. From our knowledge on the symmetry group of this equation, we obtain particular solutions and the corresponding submanifolds
X Preface
of constant curvature. This is illustrated with the classification of the toroidal submanifolds Mn of R2n-t of constant sectional curvature -1. We also classify the flat n-dimensional toroidal submanifold of the unit sphere S2n-l. We conclude this eha.pt.er hy obtaining the geometric properties of the suhma.nifolds which cor respond to the soLutions of the Generating Intrinsic equation which are invariant under (n -1)-subgroups of the group of translations.
Chapter VI describes a generalization of the Laplace method we considered in the first chapter. We introduce a Laplace transformation for Carta.n submanifolds Mn C R2n and we characterize such submauifolds for which one of its Laplace tra.n!':fot·ms l'Prluccs to a curvP. Ry considPrine; t.hf' analytic int.erpret.ation of thesf' results, we then obtain a l.ransformat.ion for overdetermined linear systems o: par tial differential equations and define its higher-dimensionaL Laplace invariants. We show that such a. system is determined in an essentially unique way by its invari ants. Moreover, we prove a fundamental theorem for the integration of systems for which the invariants vanish. We conclude this chapter by applying the theory to obtain conserved densities for systems of hydrodynamic type.
Concrete examples are provided for most of the theo:·y. In order to keep this book at a reasonable size, some topics and a few detailed proofs have been omitted. The interested reader should look up the references for more details. This book is a revised version of the lecture notes [T4] given in a short course at the Internat.ional Conference on Differential Geometry held at JMPA, Rio de Janeiro, in July 1996. Most of the nonclassical theory presented in this book, is based on research work co-authored with my collaborators M. Ablowitz, .J.L.M. Barbosa, R. Beals, P.T. Campos, J. Cavalcante, S.S. Chern, W. Ferreira., N. Kamran, M. Rabelo, C.L. Terng, and P. Winternitz. I would also like to enphasize the influence of Professor S.S. Chern who motivated my interest in this topic. The references are certainly incomplete, specially with respect to 2-dimensiona.l theory which has an extensive lit.era.t.nre.
Finally, I would like to thank Tania M. S. Sertii.o fo~ her efficency in typing most of t.he text and my son Leo for his invaluable assistance in rendering the figures contained in this book. I am grateful also to CNPq for the support given to me during the preparation of this book.
January, 1998 Keti Tenenblat
Chapter I 'Iransformations of Surfaces and Applications
In this chapter, we shall study transformations of surfaces and its applications to partial differential equations. We will express certain geometric properties of surfaces in terms of differential equations, study the transformations which pre serve the geomebc properties and we will consider the analytic formulation of these transformations in terms of solutions of the differential equations. Although most of the material presented in this chapter is classical and it can be found for example in [Bi4,Da,Ei2,Tl], we believe that starting with the presentation of the classical theory is both motivating and instructive for our later study in higher dimensions.
For this chapter, we will assume the reader has a basic knowledge of the theory of surfaces and its intrinsic geometry. The theory of moving frames will be used in most of this chapter. We have included a brief review of this theory in section 1. More details can be found in any standard text of differential geometry, for instance [O,Sp].
The goal of section 2 is to answer the following question: "What are the can nonical underlying differential equations for linear Weingarten surfaces in semi Riemannian 3-dimensional space forms'? ''. Starting in section 3 of this chapter, we will consider only Riemannian surfaces, however in this section by studying semi-Riemannian surfaces we will show that the index of the metric on the surface affects the ellipticity or hyperbolicity of the differential equations associated to the linear Weingarten surfaces. We will prove in Theorem 2.5 that such a surface, when it is parametrized by lines of curvatures, is locally determined by its metric which corresponds to a solution of one of the following differential equations: the elliptic or hyperbolic sine-Gordon, sinh-Gordon, cosh-Gordon, or Liouville equa tion, the wave equation, or the Laplace equation. In particular, we will see that Riemannian surfaces of constant Gaussian curvature correspond to solutions of the sine-Gordon equation, the wave equation, the elliptic sinh-Gordon equation or the Laplace equation.
In section 3, we shall introduce the basic concept of a geodesic congruence in a space form. Without loss of generality, we will consider the space form to be the Euclidean space R3 , the unit sphere S 3 or the 3-dimensional hyperbolic space H3• We will start with the simplest examples of transformations of surfaces by considering parallel surfaces. Bonnet's theorem in a space form will be given, re lating surfaces of positive Gaussian curvature parallel to surfaces of constant mean curvature. We will show that linear Weingarten surfaces lie in a one-parameter family of parallel surfaces that include at least one minimal surface or a surface of
2 I. Transformations of surfaces and applications
constant Gaussian curvature.
Section 4 contains the classical results on transformations of surfaces in R3 of constant negative curvature, obtained by Backlund and Bianchi [Bal,Ba2,Bil,Bi2]. \Ve will introduce the pseudo-spherical geodesic congruence~ in a space form of cur vature J{ and we will prove Backlund's theorem in a space form. This theorem gives a transformation between surfaces of the same constant Gaussian curvature K, I< < I<. The composition of such transformations will be considered in the Permutability theorem of Bianchi. The reader is referred to [T5) for a computer animated video which provides illustrative examples of the geometric theory pre sented in sections 3 and 4.
The analytic interpretation of the theory presented in section 4 will be given in section 5 in terms of solutions for the sine-Gordon equation. We will obtain the so called Backlund transformation for this equation. This transformation provides a method of obtaining, from a given solution of the sine-Gordon equation, a new solution of the same equation. The analytic interpretation of the Permutability theorem will give the superposition formula which enables one to obtain more solutions of the sine-Gordon equation algebraically. In section 5, we shall also obtain the Backlund transformation and the Superposition formula for the elliptic sinh-Gordon equation. The proofs in this case will be purely analytic.
In section 6, we shall consider the classical Laplace transformation for second order hyperbolic equations and the method of solving such equations by using their Laplace invariants (see [Da,Gou]). The geometrical interpretation of this theory will be given by transformations of surfaces in Euclidean space which generically preserve the property of being parametrized by conjugate curves.
In section 7, we will introduce the notion of differential equations which de scribe pseudo-spherical surfaces. The study of such equations was introduced in [ChT2] and it was considered in subsequent papers [CavT,JT,KTl,Ra]. We will show that, besides ~he sine-Gordon equation given in §2, there are other diffe rential equations, such as the Korteweg-de Vries, the modified Korteweg-de Vries, the Burgers equation, etc. whose generic solutions are in correspondence with 2-dimensional Riemannian manifolds with constant negative curvature. As an ap plication, we will show how the geometric theory of such manifolds can be used in order to obtain Backlund transformations and conservation laws for the differential equations.
§1. The structure equations.
We will denote by M; a semi-Riemannian manifold of index r and by M~(K) a simply connected semi-RJemannian manifold of constant sectional curvature K of index s. Let M; be isometrically immersed in M~(K), where r :5 s. We consider
1. The structure equations 3
an adapted frame e1, e2, e3 in M such that el> e2 are tangent toM and
1 ~I, J ~ 3
where a;= 1, 1 ~ i ~ 2, except for r indices where a;= -1, and a3 = 1 if s =,.. and a3 = -1 if s < r.
Let WJ be the dual forms and WJJ the connection forms defined by
3
where
3
3
niJ = -KalaJWri\WJ.
Tf WP. rP.stric:t those forms toM we have w3 = 0 and we obtain from (1.2),
2
2
2
(1.1)
(1.2)
( 1.3)
(1.4)
(1.5)
(1.6)
The Gauss and Codazzi equations follow from ( 1.3) and arc given respectively by
2
(1.7)
(1.8)
Since dw12 is a 2-form, it is a multiple of w1 /1. w2. Hence we define the Gaussian curvature /( of M, by
(1.9)
The mean curvature H of M is defined by
(1.10)
hence Wt 1\ <T2W23 + <TtWt3 1\ W2 = 2Hwt 1\ W2. (1.11)
Moreover, it follows from (1.7) and (1.8) that
(1.12)
Observe that (l.l) and (1.6) imply that de3 = - L<T;b;jWje;. Therefore the ij
matrix associated to the linear map de~j is diagonalizable, if and only if, at each point we have
(o-tbn + o-2b22f2- 4uta-2(bub22- bi2) ~ 0,
i.e. H 2 - a3(K- I<) ~ 0. Tf Wf'. want thP. P-igenva.lue.s to he distinct, we need the surface to satisfy the property
( 1.13)
In this case, we may consider e1, ez to be eigenvectors a.t each point. Hence the connection forms w;3 are of the form w;3 = h;w;. We say that <T;h; are the principal curvatures of M and e1, e2 are the principal directions.
The first fundamental form at a point p E M is defined by
where T.,M is the tangent plane of M at p and ( , ) denotes the metric on M induced by M. The second fundamentai for·m is defined by
If X(x1, x2 ) is a loca.l parametrization of Jl.f C M whose coordinate curves are lines of curvature, we may consider Xx, = g;e;, 1 :<:; i :<:; 2. Therefore, se get
I <Tzgfd:ci + a2g~dx~ I I = h 1g~dxi + h2g~dx~
(1.14)
§2. Differential equations associated to linear Weingarten surfaces.
In this section, we shall obtain the cannonical differential equations which corre spond to linear Weingarten surfaces immersed into a semi-Riemannian space form.
2. Differential equations associated to linear Weingarten surfaces .j
We will show that such a surface, when it is parametrized by lines of curvature, is locally determined only in terms of its intrinsic geometry, namely its first fun damental form wh:ch is in correspondence with solutions of certain differential equations, which are elliptic or hyperbolic according to the index of the metric. In particular, surfaces of constant Gaussian curvature immersed in Riemannian space forms are shown to correspond to solutions of the sine-Gordo:~. equation, the wave equation, the elliptic sinh-Gordon equation o~ the Laplace equation. Classi cal results relating Weingarten surfaces in 1f3 to the sine-Gordon equation can be found in Darboux [Da Vol III]. In Euclidean and Minkowski spaces, such surfaces have been studied more recently in [Bul,Wu,M,R].
We start by recalling the definition of a Weingarten surface. We will consider semi-Riemannian surfaces M; of index r, contained in a semi-Riemannian space
form M~(I<.); of index s, r ::::; s and constant Gaussian curvature /\.
A surface M contained in a space form M~( K) is a Weingarten surface if there exists a differentiable function relating the mean curvature H and the Gaussian curvalure K uf M. A surface is called a linear Weingar·ten surface if the mean curvature and the Gaussian curvature satisfy a linear relation.
In this section, we will consider semi-Riemannian Weingarten surfaces M; con
tained in M!( K), which satisfy a relation of the form
a+ 2/3H + a3"Y(K- K) = 0, (2.1)
where a3 = ±1 is the length of the unit vector field, normal to the surface and o:,{), "Y are constants such that
We are also assuming, K- [{ ::/: 0. Such a surface will be called a linear Weingarten
surface in the space form Nl. Minimal surfaces ( a="'= 0, /3-::/: O) a.nd surfaces of constant nonzero mean curvature ( 1' = 0, a # 0, /3 f O) or constant Gaussian curvature (!3 = 0, ~( f 0) are particular cases of linear Weingarten surfaces.
Assuming that M; has two distinct principal curvatures at each point, we will show (Theorems 2.5 and 2.10) that, if "Y f 0, such a surface corresponds to a solution 'l/J(x11 x2 ) of a differential equation of one of the following types:
{ sin(c+l/1) a2'¢':c:1X1 - Clt?/Jr2X2 = 0
or
6 I. Transformations of surfaces and applica.tjons
where a1 and a2, which are equal to ±1, depend on the index of the metric on the surface and c is a real number determined by the constants a, /3, {, 1\.
If b = 0 then the surface has constant mean curvature H (in particular it is a minimal surface when H = 0) and it corresponds (Theorem 2.8) to solutions of the following differential equation
a _ { a3H 2 (e- 2"'- e21#)- /{e2tJ! if H =f. 0, a! '1/Jx,x, + 2 ,P"'2x~ - a3C 2e-2t/l - I\ e2"' C' =f. 0, C E R, if H = 0.
The right hand side of this equation can be reduced to a multiple of sinh( c ± 2'1/.' ), cosh(c ± 21/-•) or e-2,;,, depending on the values of a 3 and I<. Before proving those results we give some explicit examples.
2.1 Example. We consider Enneper's surface
xi 2 x~ 2 2 2) X(x 1, x2 ) = (x1- 3 + x1x2, x2- 3 + Xz:l:p x 1 - x2
Fig. 2.1 Enneper's minimal surface.
which is a minimal surface in R3 , parametrized by lines of curvature. It is easy to see that the first fundamental form is given by
I= (I+ X~+ x~?(dx~ + dxn.
We define a function li•(.r., x 2 ) by
Then '!jJ is a solution of the differential equation
We observe that by rescaling the independent variables we can change the constant 4 that appears multiplying the right hand side of this equation and reduce it to 1.
2. Differential equations associaied to linear Weingarten surfaces 7
2.2 Example. The pseudo-sphere is a surface of R3 described by
X(x 11 x2) = (sech x,cosx2 , sech x1 sinx2 , tanhx 1 - xt),
Its Gaussian curvature is constant ]{ = -1 and its first fundamental form is given by
We consider a function t/.•, determined by the coefficients of I, defined by
t/J cos 2 = tanh x 1
t/J sin 2 = sech x 1•
Then t/J is a solution of the differential equation
Fig. 2.2 The pseudo-sphere.
In what follows we will obtain the differential equations whose solutions are in correspondence with the linear Weingarten surfaces. The proofs will also show how to define the function t/J associated to a linear Weingarten surface. More examples will be given later. We will consider three cases for the constants a w_d '"Y of the linear relation (2.1 ), namely: ~ #: 0, '"Y = 0 and '"Y #: 0 with a = 0. In each case the Weingarten surface will be characterized in terms of solutions of nonlinear differential equations, assuming that a.t each point of the surface tht>re are two real distinct principal curvatures.
We start with the case a'"Y #: 0. Without loss of generality, we may consider a=l.
2.3 Lemma. Let M; C M~(I<) be a surface which satisfies
Assume ( 1.13) holds, and let
T; = 1 + 2cr;{3h; + '"Yh? 1:::; i:::; 2.
where cr;hi are are the principal curvatures. Then
D(u2h2- u1h1) 2 ,
= 7(u2h2- u1h1 ) 2 •
Moreover, if D = 'Y- (J2 > 0 then T; > 0,1 ~ i ~ 2.
Proof. Since (1.13) holds, the principal curvatures u;h;, 1 ~ i ~ 2 exist and satisfy the following relation:
1 + /3( a1h1 + a2h2) + "'((11a2h1h2 = 0,
which follows from (2.2) and (1.12). From (2.5) we get
h2 = _ 1 + (3u1h1 ht = 1 + (3a2h8
tTzfJ + "ftTt a2h1 ai{3 + 'Ya1a2h2 ·
Therefore, the difference of the principal curvatures can be expressed as
(2.5)
(2.6)
(2. 7)
If follows that (2.4) holds. Moreover, since T; = (1 + u;(3h;)2 + Dh~, we conclude that 'Ji > U whenever D > U.
0
Our next result characterizes a linear Weingarten surface by its first fundamen tal form.
In order to state the theorem, we need to introduce some notation. Given a linear Weingarten surface satisfying (2.2) we define the constants
A= f.2UtU3f3/!Dih2 B = (11(12[-K + UJ('y- 2f32)/'Y2}
L =(I- a3K'Y)2 + 4u3 K(32 ,
·(2.8)
where D = "(- (32 and E:2 = ±1.
2.4 Theorem. Let M; C M~(K) be a linear Weingarten surface which satisfies (2.2), where 'Y =f: 0. Assume that a.t each point there are two real distinct principal curvatures, then there exist local coordinates x 1 , x 2 such that the first fundamental form is given by
where g~ + tgi = f:y , f = ±1'
(2.9)
(2.10)
(2.11)
(2.12)
Conversely, suppose that a pseudo-riemannian surface can be parametrized by local coordinates such that the first fundamental form satisfies (2.9}- (2.12}, then there
2. Differential equations associated to linear Weingarten surfaces 9
exists a linear Weingarten surface in a semi-Riemannian manifold M 3(K) which satisfies a relation of the form (2.2}, whose first quadratic form is given by {2.9} and its second fundamental form is
(2.13)
where
(2.14)
e:i = t:~ = 1 and t:1t:2 = -u1u 2t:, f = si9nD, D = -y- /32•
Proof. M; is isometrically immersed in M;( K) therefore, we may consider an adapted frame e1, e2, e3 such that e1 , e2 are tangent to M and
where u17 u2 are equal to 1 except for r indices from 1 to 2, where u; = -1. Moreover, U3 = 1 if s = r and u3 = -1 if s < r.
By hypothesis H 2 - u3 ( K - K) > 0, therefore we may choose the frame e1, e2
such that Wt3 = htWt
We define functions 9i, 1 S i S 2 by
1 2=>..;T; 9;
where 1i is given by (2.3) and A; = ±1 so that >..;T; is positive. We will show that
d ( ;,i) = 0 ' 1 s i s 2.
In fact,
From (2.15) we have
(2.15)
(2.16)
It follows from the Codazzi equations (1.8) and the structure equations (1.5) that
dh; 1\ w; + h;dw; = u1u2hjdw; where 1 S i :f:. j :::; 2.
'I'herefore, substituting these equations into (2.16) we obtain
From (2.6) we have that -hi+ u;u;h: = -Ti/({3 + '"(O';hi), hence we conclude that d(w;fg,) = 0. Therefore, on a.n open contractible region U of M there exist smoth real valued functions X11X2 such that dx; = w;fg;, i.e., we may consider the frame ei = Xz./9i· Since
Wt = 9tdXt 1 W2 = !J2dX?. 1 (2.17)
the connection fqrm w12 is determined by the structure equations (1.5) as
(2.18)
and we have (2.19)
Hence the fundamental forms are gi,•en by (2.9). Substituting (2.18) and (2.19) into the Coda.zzi equations (1.8) we obtain
which reduce to
h1,:1:291 + ht91,;z;2 - O't0'29l,Z2h2
h2,:r1Y2 + h2Y2,z1 = 0'1 0'292,:&1 ht,
By hypothesis M 2 is a. linear Weingarten surface, therefore from (2. 7) w~ get
!la log(g~IT,I) = o, UX2
!la log(g~IT21) = o. UXt
Since (2.4) holds and T; > 0 for 1 ~ i ~ 2, without loss of generality we may consider T1 > 0. It follows that we can change the variables x 1 and x2 separately such tha.t
(2.20)
where e; = signD. As a consequence of (2.5) it follows that (2.4) is satisfied. From (2.4) and (2.20) we conclude that (2.10) is verified.
It follows from the Gauss equation (1.7), (2.17) a.nd (2.18) that
( (gz ).,1 ) ( (Yt )"'2) ( r.' h h ) 172 -- + O't -- = - .l\ O't0'2 + O'J 1 2 9192·
91 .,, 92 .,, (2.21)
Now we want to obtain h1h2 in terms of g1 and !J2, in order to reduce (2.21) to a differential equation for g1 and !/2· Now, from (2.20) and (2.10) we have that
2. DiffereiiLial equatiuw; associated to linear Weinganen surfaces 11
h1 and h2 satisfy (2.14) where e~ = e~ = 1. By substituting (2.14) into (2.5) we obtain
(2.22)
(2.23)
It follows that (2.21) reduces to (2.11) where A and B are constants defined as in (2.8). It is easy to check that the first relation of (2.12) holds. Moreover, 1- 4c::A2b2 = (b- 2a2 ) 2 fb2 ~ 0.
Conversely, suppose a surface M 2 can be parametrized by local coordinates such that (2.9) - (2.12) hold. We fix o-3 such that o-~ = 1 and we define the constant ,8 ~ 0 by
(2.24)
where the sign in front of the square root can be positive or negative if c = 1 and it is the sign of "' if c:: = -1. This implies that si9n(1' - ,82) = c
Now we choose c::2 = ±1 so that
(2.25)
(2.26)
We consider the second quadratic form as in (2.13) where we define h1 and h2 by (2.14).
We claim that these quadratic forms satisfy the Gauss and Codazzi equations
of an immersed Weingarten surfaze IV!'; in M~( K) where r is the number of indices 1 :S i :S 2 for which o-; = -1, and sis the total number of indices O"I, 1 ~ I~ 3 for which 0"[ = -1. Moreover, the surface satisfies the relation (2.2).
In order to prove that the Gauss equation is satisfied, it follows from (2.11) that we only need to show that
-- 2 2 -o-3h191h292- Kq1o-29I9z = A(gl- eg2) + B9192·
'I'his is an easy computation that follows from (2.24)- (2.26) and (2.14). To prove the Codazzi equations we need to show that
(h1gt),:e2 = 0"10"2 9t,z2h2,
(h292),xt = 0"10"2 92,xth1
12 I. Ttansformations of surfaces and applications
Therefore, the fundamental theorem for submanifolds of M~(K) implies that there exists a stu·face M;, locally defined, with the given first and second funda mental forms.
Now using (2.14) we compute t.hP. Gaussian and the mean curvaturP. of M by
using formulas (1.10) and (1.12). A simple computation now shows that
1 + 2aH + a3b(J<- K) = 0.
0
As a consequence of the theorem above we have characterized a linear Wein garten surface by its metric, which is given by the solutions of equations (2.10) (2.12). Our next step consists in rewriting those equations in terms of a single function t/1( x1x2).
2.5 Theorem. Let M; C M~(K) be a linear Weingarten surface 'Which satisfies (2.2) 'Where 1 =J= 0. As.~ume that at each point there are t?llo di.~tind principal curvatures. Then there exist local coordinates Xt. x2 for M 2, a function tP( x 1 x2) and constants A, B and L defined by (2.8) such that (i) If D > 0, then 1 > 0, L 2:: 0 and 'ljJ satisfies the differential equations
0'2tPrtr1 - O't t/Jr2r2 = .JL sin( C + tP)
where c is a constant defined by
A = y'L sine 21' '
( i i) If D < 0, then t/1 satisfies the equation
u,,P.,., + u,.p.,., = {
vi sinh(c ± t/1)
±y'iLI cosh( c ± '¢)
(2.27)
(2.28)
(2.29)
(2.30)
and the sign in (2.29), (2.30} is equal to the sign of B for L > 0 and to the sign of A for L < 0.
Cunversely, for any solution "'(;qx2) of (!2.27) or ( 2.29) there exists a linear
Weingarten surface M; C M~(K) satisfying (2.2) whose first and second funda mental forms are determined by'¢. More precisely, if 1/J satisfies (2.27) thf first
2. Differential equations ~:sociated to linear Weingaden sudaces 13
fundamental form of M is given by (2.9} wherw
1/J 91 = .,ncos 2'
. f/j 92 = ,;;y Sill 2' (2.31)
1 > 0 and the constants L, A, B satisfy (2.8}, (2.12} with t: = 1 and (2.28}. If '¢• satisfies (2.29}, the first fundamental form is given by (2.9) where
gt = yT-Yi cosh %, 92 = yT-Yi sinh ~ if')'< 0,
(2.32)
9t = .,nsinh t• t/J 92 = .,ncosh 2" if')'> 0.
and the constants satisfy (2.8), (2.12) with co = -1 and (2.30) if L # 0. In both cases, the second fundamental form is defined by (2.13) and (2.14).
Proof. In Theorem 2.4 a linear Weingarten surface is characterized by its first fundamental form which is given by (2.9), where g1 and 92 satisfy (2.10)- (2.12).
i) If t: = 1, then from (2.10) and : 2.12) we have 1 > 0 and L ~ 0. Therefore, we may consider 91 and 92 defined as in {2.31). Let c be a constant defined by {2.28) then the differential equation {2.11) reduces to (2.27).
ii) If t: = -1, since 1 # 0, it follows from (2.10) that we may consider 91 and 92 defined by (2.32). Hence in both cases equation (2.11) reduces to
(2.33)
Now we need to consider three cases, namely L > 0, L < 0 and L = 0. tor L =/= 0, let c be a constant defined by (2.30) according to the sign of B for L < 0 and to the sign of A for L > 0. Then (2.33) reduces to the first and second equations of (2.29) depending on the sign of L. If L = 0, it follows from (2.12) that B = ±2A. Therefore, (2.33) reduces to the third equation of (2.29). This concludes the proof of Theorem 2.5.
2.6 Example. We consider the family of surfaces in R3 parametrized by
X(xt. x2) = (sech Xt cos x2, sech X1 sin x2, x1 - sech x1sinh xi)
a(sech x1cosx2sinh xx, sech x1 sinx2sinh x1 , sech x1)
D
where a is a real constant. For each nonzero value of a, the surface described by X (see Fig. 2.3) is at a constant distance a from the pseudo-sphere (see §3 for the
14 1. Transformations of surfaces and applications
notion of parallel surfaces and Example 3.8). Each surface of this ramily, off its singular points, is a linear Weingarten surface which satisfies the relation
1+2aH+(l +a2)K=0.
a=O a=3 a=7
a= 12 a= 19
Fig. 2.3 Linear Weingarten surfaces which satisfy the relation 1 + 2aH + (1 + a2 )K = 0.
2. DiffereutiaJ ~quatiuns associated to linear Weingarten liurfaces 15
The set of singularities of each surface is generated by (x 1 ,x2) such that
(a +sinh.rt)(l- asinhx1) = 0.
The first fundamental form of the surface X(x 1 , x2) is given by
I = secb2 x1 ((a+ sinh xddxi + (1 - asinh xddxn.
Following Theorem 2.5 we define
cos~ = sech x 1(a +sinh xt) sin~ = sech x1( 1- asinh xt).
Then t/J is a solution of the differential equation
tPx1x1 - tPx2x2 = 1 ~ a2 sin(~'+ c).
where cis a constant. We observe that whenever a= 0, the surface X reduces to the pseudo-sphere (see example 2.2).
We observe that in Theorem 2.5 whenever the constants which appear mul tiplying the right hand side of the equations (2.27) and (2.29) are nonzero, they can be reduced to ±1, by rescaling the independent variables. Then the equations will be of the form ( * ), proving that the linear Weingarten surfaces are in corre spondence with solutions of the elliptic or hyperbolic sine-Gordon, sinh-Gordon, cosh-Gordon or Liouville equa.Liou, the wave equation or the Laplace equation. Moreover, in the Riemannian case (s = r = 0), when the constant f3 vanishes, the surface has constant Gaussian curvature K and Theorem 2.5 reduces to our next corollary. Namely, if R: < K (resp. K > K), then equation (2.27) (resp. (2.29)) reduces to the sine-Gordon equation (resp. the elliptic sinh-Gordon equation) for K =/= 0 or the wave equation (resp. Laplace equation) for K = 0.
2.7 Corollary. Let M 2(K) C M 3(K) be a surface of constant Gaussian curvature K contained in a Riemannian 3-dimensional space form such that K # K. If K > K, assume M has no umbilic points. Then there exist local coordinates Xt, x2 , and a function t/J( Xt, x 2) which satisfies the differential equation
w,,z, - 1/;.,2"'• = -I< sin t/J t/J., 1.,1 + .,P.,2 x 2 = -K sinh t/J
if K < K,
if K >I<.
(2.34)
(2.35)
Conversely, suppose t/J is a solution of (2.34) ( resp. (2.35}}. Then there exists
a surface of constant Gaussia'.!._ curvature K in a space form M 3(K), which is
unique up to rigid motion of M, whose first and second fundamental forms are given respectively by
1={ cos2 ~dx~ + sin2 ¥dx~ cosh2 ~dx~ + sinh2 ~dx~
if if
I. Transformations of surfaces and applications
VIK- Kl sin~ cos *(dxi- dx~) VIK- Kl sinh* cosh *(dxi + dx~)
if
if
K>K. (2.37)
Proof. The proof follows from Theorem 2.5. The surface M satisfies the linear relation (2.2) where 0'1 = 0'2 = 0'3 = 1,13 = 0 and D = 1 '# 0, i.e. l+y(l\ -K) = 0. From (2.8) we get the constants A= 0, B = -K and L = (1- K1)2 2: 0.
If K < K, then 1 > 0 and we apply Theorem 2.5 i). It follows that there exists local coordinates XI. x2 and a function 11'( x11 x 2 ) which satisfies
where we used (2.28) to obtain VI= -1 K. Conversely, if 1/· satisfies this equation
there exists a surface M(I{) C M(K) whose fundamental forms are determined by ,P as in Theorem 2.5. By scaling the variables x 1 , x2 by vft we conclude the proof for the case K < K.
When K > I<, the proof follows similarly by applying Theorem 2.5 ii) where L 2: 0. In fact, L = 0 if and only if K = 0. In this case, (2.35) agrees with the third equation of (2.29). If L > 0, then it follows from (2.29) and (2.30) that v'L = =t=biK and ·I/Jx1x1 + t/J:c2x 2 ==Filii< sinh(±,P ). By rescaling x1 and x2 we get (2.35).
0
The Corollary above shows that surfaces of constant curvature K contained in a three-dimensional space form of curvature K, K '# K, are determined by solutions of equations (2.34) and (2.35). In particular, surfaces of constant neg ative curvature (hyperbolic surfaces) correspond to solutions of the sine-Gordon equation V':c1x 1 - I/Jr2:c2 = sin tj.>. Hilbert's theorem asserts that there are no com plete hyperbolic surfaces in R3 . This is a consequence of the that the sine-Gordon equation does not have a solution 1/J defined on R2 such that 0 < 1/J < 1r (see for example [Sp]).
The linear Weingarten surfaces which satisfy (2.1) with 1 = 0 correspond to surfaces of constant mean curvature II. In this case we have the following result.
2.8 Theorem. Let Af; C M~(I<) be a surface of constant mean curvature H. Assume that at each point there are two distinct principal curvatures, then there exist Local coordinates x 1,x2, and a function 1jJ(x1ox2) such that
a} If H # 0, 'If satisfies
(2.38)
b) If H = 0, t/J satisfies
(2.39)
where C is a non zero constant.
Conversely, fo·r any solution ,P(xb x2) of (2.38} {resp ( 2.39)} there exists a surface
of constant mean curvature H (resp. a minimal surface) M; C M~(K), whose fundamental forms are given by (1.14) where 9t.92 are defined by
and h 1 , h2 are defined by
92 _ 92 _ e2"' I- 2-
h1 = O't(Ce-2"' +H), h2 = u2(-ce-2"' +H),
where C = IHI for H ;f:. 0.
Proof. The arguments of the proof are similar to those used in the proof of Theorem 2.5. Let T; be the function given by
1i = -2(H- u;h;); 1 ~ i ::; 2, (2.40)
where u;h; are the principal curvatures. Since T1 T2 = -( u2h2 - u1h1 )2 < 0, we may consider, without loss of generality, T1 > 0 and T2 < 0. By defining the functions 91 and 92 as
(2.41)
we show the existence of local coordinates x 1 , x2 such that the first and second fundamental forms are given by (1.14). Moreover, we have
f) -8 (log g~T2) = 0
XI
Hence, we may change variables x 1 , x2 separately such that
(2.42)
where C > 0 is a constant. Since T1 + T2 = 0 we get 9~ - 9~ = 0. Let t/J( u, v) be a. function such that
g~ = 9~ = e2"'. (2.43)
h1 = u1(Ce-2"' +H), h2 = u2(-Ce-2.P +H) (2.44)
and the Gauss equation reduces to
0'1tPz1:::1 + 0'2tP:::,:z:• = U3C2e-2.P- (K + 0'3H2 )e2"'. (2.45)
In particular, when H = 0 this equation reduces to (2.39) and whenever H i= 0 we can take the constant C = IHI and (2.45) reduces to equation (2.38).
The converse follows by defining the first and second fundamental forms of the surface as in ( 1.14) where 9t. 92 are given by (2.43) and h1 , h2 are defined by (2.44) where C = IHI for H =F 0.
Cl
Example 2.1 illustrates Theorem 2.8 b). The tori with constant mean curvature immersed in R3 , discovered first by H. Wente [We], correspond to special doubly periodic solutions of the sinh-Gordon equation (see also [Abr]).
2.9 Example. Let t/J(xt) be a solution of equation (2.38) where 0'1 = 0'2 = ua = 1, H = 1 and I< = 0 i.e.
such that '¢(0) = 0 and '¢,1 (0) = m/2. Define the function
Then
Xt > 0,
has constant mean curvature H = 1 and X is an isothermal representation of the rotational surface of Delaunay.
In the next theorem we consider linear Weingarten surfaces M~ C M~(K), which satisfy a relation ofthe form {2.1) where -y i= 0 and a= 0 i.e. D = -/32 i: 0. The proof in this case follows the same arguments used in Theorem 2.4.
2.10 Theorem. Let M: c M!(K) be a linear Weingarten surface which satisfies
2/3H + -yu3(I<- K) = 0. (2.46)
where -y/3 =/= 0. Assume that at each point there are two distinct principal cur vatures. Then there exist local coordinates x~,x:~ for M 2, a function '¢(x~,x2), constants A, B and L = K 2-y2 + 4u3K/32 such that
{ v'l sinh( c ± ¢) if L > 0
0'2tPx1x 1 + O'J''P:c2x 2 = ±y'iLi cosh(c ± t/J) if L < 0
2Ahle2"1' if L = 0
where c is a constant defined by
{ ~sinh c if L > 0
A = ± ~cosh c if L < 0
B = { ±~cosh c if L > 0
v'iLi sinh c if L < 0 IT
(2.48)
and the sign in (2.47}, (2.48} is equal to the sign of B for L > 0 and the sign of A for L < 0. Conversely, for any solution ¢(xtx2) of (2.47) there exists a Weingarten surface
M; C M~(K) satisfying (2.46}, whose first and second fundamental forms are given by (1.14}, where
g1 = VI-Yf cosh t 91 = /GI sinh t
92 = VI-Yf sinh t if r < 0
92 = /GI cosh% if 1 > 0.
and the constants satisfy (2.48} and B 2 - (2A)2 = Lh2 . Moreover,
where t:~ = f~ = 1 and
Proof. We define the functions
(2.49)
(2.50)
(2.51)
(2.52)
where o-;h; are the principal curvatures. Since T1T2 = -[J2(o-2h2 - o-1h1)2 < 0, we may consider w.l.o.g. T1 > 0 and T2 < 0. By considering the functions 9t.92
defined as 1 2=T1 gl
we show the existence of local coordinates x1, x2 such that the first and second fundamental forms are given by (1.14). Moreover,
Hence, we may change the variables Xt and x2 separately such that
9iT1 = [32 and g~T2 = -[32
Claim: g1 and 92 satisfy
(2.53)
(2.54)
(2.55)
where
(2.56)
and
L = K2 12 + 4a3K ~2
In fact, equation (2.54) follows from (2.52) and C 2.53) while, (2.55)-(2.57) follow from the Gauss equation and the expressions of h1 and h2 in terms of g1 , 92 obtained by using (2.52) (2.53) and (2.54). The expression of h1 and h2 are precisely given by (2.50) and (2.51).
Conversely, for each pair of differentiable functions 91 ,92 satisfying (2 . .54 )-(2.56)
there exists a linear Weingarten surface in a semi-Riemannian manifold M3 (K) which satisfies a relation of the form (2.46).
In fact, for a fixed <13 such that <1~ = 1, we consider f 2 = ±1 such that
Now we define the constants a > 0 and K by
and the functions h1 and h2 by (2.50) and (2.51). It follows that the quadratic
forms (1.14) satisfy the Gauss and Codazzi equations of a. surface M; C M~(I<). Moreover, equation (2.46) is satisfied.
The next step consists in rewriting (2.54)-{2.56) in terms of a function 1/'(Xt, x2). Since "Y '=? 0, it follows from (2.54) that we may consider 91 and 92 defined by (2.49). Hence in both cases equation (2.55) reduces to
By considering the cases L > 0, L < 0 and L = 0 we conclude that this equation reduces to (2.47) and (2.48).
0
We observe that the nonzero factors which appear on the right hand side of equation (2.47) can be reduced to ±1, by rescaling the independent variables.
2. Diffen1IJtial equations associated to linear Weingarten surfaces 21
2.11 Example. We consider the family of Weingarten surfaces in R3 parametrized
by 3 3
( XI 2 X2 2 2 2) X(xh x2) Xt - 3 + XtX 2, x2- 3 + x2x 1, x.- x2
+ a( -2Xt. 2x2, 1 -xi- xi} I (1 +xi+ xn.
where a is a real constant (see Fig.2.4 and also Example 3.8b in §3).
H + 12K = 0
H+2.5K=O
H=O
H-2.5K=O
Fig. 2.4 Linear Weingarten surfaces which satisfy the relation H- aK = 0.
Each surface X, off its singular points, satisfies the relation H - aK = 0. The set of singularities corresponds to (xh x2) such that (1 +x? + xD4 = 4a2. The first fundamental form of t.lte surface :r is given by
( ?a) 2 ( 2a)2 I= f- -e dx~ + e + f dx~,
where
f = 1 +X~+ X~.
Following Theorem 2.10, when a=/: 0 we define a function t/J by equation (2.49) i.e.
rn 1/.1 2a 2y .!:COSh - = f + -
2 e 2v'2sinh! = e- 2a. 2 e
Then 1/J is a solution of the differential equation
When a = 0 the surface X reduces to the Enneper surface and it was considered in Example 2.1.
In this section, we have seen that linear Weingarten surfaces are in corre spondence with solutions of certain differential equations, some of them highly nonlinear. In section 7, we will see that there are other nonlinear differential equations such as the Korteweg-de-Vries (KdV) equation, the modified Korteweg de-Vries (MKdV) equation, Burgers equation, etc. whose solutions are associa.ted to surfaces of constant negative Gaussian curvature.
§3. Geodesic congruences and parallel surfaces
In this section, we shall begin the study of geodesic congruences in a Riemannian
3-dimensional space form M 3 ( K) and pairs of surfaces in M related by such a
congruence. In pa1·ticular, we will consider parallel surfaces in M and we will show how the mean and Gaussian curvatures of such surfaces are related to each other. As a consequence, we shall see that surfaces of nonzero constant mean curvature in M are parallel to surfaces of constant positive Gaussian curvature (Corollary 3.4). This result is known as Bonnet's theorem when the ambient space is Euclidean. Moreover1 we will prove that a surface which is parallel to a linear Weingarten surface is also a linear Weingarten surface and in particular, flat surfaces are parallel to flat surfaces. We will conclude this section by showing that generically a linear Wcingn.rtcn surface is locally parallel to a minimal surface or to a surface of constant Gaussian curvature. We will start by defining a geodesic congruence in a space form.
3. Geodesic congruences anu parallel surfaces 23
A geodesic congruence in M 3 ( K) is a 2-parameter family of geodesics in M. Without loss of generality we consider /{ = 0, 1 or -1. We denote by S3 the unit sphere in Jl4 and by H 3 the hyperbolic space characterized as the subspace of the Lorentzian space L4 determined by the vectors of length -1.
Locally, a geodesic congruence is given by
cos A X(XtX2) +sin). ~(XtX2) if M = S3 c R4
cosh .X X(x1x2) +sinh.,\ ~(x 1 x2) if M3 = H3 C L4
where X: M2 -+ .n.'f3 is an isometric local embedding of M2 in M3,~(x 1 x2 ) is a
unit vector tangent to M at X(x 1x2) and A E I CR. Such a geodesic congruence is said to be normal to M, if ~(p) is normal to M
at p, for each p E M and the congruence is said to be tangent to M if ~(p) is tangent to M at p, for each p E M.
Let M and M' be two surfaces isometrically immersed in the space form M 3
and let l : M -+ M' be a diffeomorphism, such that for each p E M and p' = f(p)
there exists a unique geodesic "' in M joining p and p'. Such a congruence will be called a geodesic congruence between M and M'.
In the remainder of this section, we will consider a special kind of geodesic con gruence which exists between parallel surfaces. Other congruences will be treated in the following section.
3.1 Example. Let M c M3(K) be an orientab!e surface and let e3 be a unit
normal field on M. We consider a surface M' C M to be parallel to M, if there is a normal geodesic congruence between M and M' such that the distance between corresponding points is constant, i.e. for each p E M we have
{ p+ae3 if M=R3
I • if M =53 (3.1) P = cos a p + sm a e3
cosh a p + sinh a e3 if M = H3 c L4
where a ::/: 0 is a real constant. We say that M and M' are parallel s·urfaces at a distance a.
Given a surface M C M 3 (K), for which values of a does p', given by (3.1), define a parallel surface? We denote by Hand K the mean and Gaussia:J. curvatures of M. Suppose that for each p EM,
1 - 2aH + a2 K #- 0 if M = RJ,
cos 2a - sen2aH + sen2aK #- 0
cosh 2a - senh2aH + senh2 K ::/: 0
if M = S3,
then (3.1) defines an immersed surface M' in M parallel to M. In fact, let X be a local parametrization of M and let et, e2, e3 be an adapted moving frame on M. We denote by wh w2 the dual forms and by Wii the connection forms. Then it follows from (3.1) that
dX' = (cos aw1 -sin aw13)e1 + (cos aw2 -sin aw23)e2, { (w1- aw13)e1 + (w2- aw23)e2, if/{= 0,
if [{ = 1,
if K = -1. (cosh aw1 - sinh aw13)e1 + (cosh aw2 - sinh aw23)e2,, (3.3)
Hence, for each case K = 0, 1 or -1 we get respectively
( Wt - aw13) 1\ ( w2 - atv23) = (1 - 2aH + a2 I<)w1 1\ w2 (3.4)
(cos aw1 -sin aw13) 1\ (cos aw2 -sin aw23) = (cos 2a- sin 2aH + sin2 al<)w1 1\ w2
(cos aw1-sinh aw13)/\( cosh aw2-sinh aw23) = (cosh 2a-sinh 2aH +sinh2 ai<)w1/\w2
Therefore, X' is a parametrized regular surface, whose normal vector at each point is parallel to e3 at the correponding point of X.
The following result relates the Gaussian and the mean curvatures of parallel surfaces.
3.2 Proposition. Let M be a regular orientable surface in M 3 ( K) and let a be a real constant such that
1 - 2aH + a 2 /{ =f. 0 if K = 0,
cos2a- sin2aH + sin2 aK =f. 0 if [( = 1, (3.5)
cosh 2a- sinh 2aH + sinh2ai< # 0 if K = -1.
Then the curvatures H' and K' of a surface M', parallel to M at a distance a are given respectively by
H'=
and
H -aK 1-2aH+a2 I< sin 2a +cos 2aH- sin a cos a/{
cos 2a - sin 2aH + sin2 aK - sinh 2a + cosh 2aH - sinh a cosh aK
cosh 2a- sinh 2aH + sinh2 al<
K 1-2aH+a2/{
cosh 2a - sinh 2aH + sinh2 aK
if K = 0,
if [{ = 1, (3.6)
if K = -1,
3. Geodesic congruences and parallel surfaces 25
Proof. I) We first prove the case in which the ambient space is Euclidean. Let et. e2, e3 be a moving frame associated to the surface M. If X is a local parametrization of M, then the parallel surface M' is parametrized by
X'= X +aea.
(3.8)
whert: Wt- aw13 and w2- aw23 are linearly independent as a consequence of (3.4). Moreover, we may consider the moving frame adapted to X', given by
e: = e;, 1 ::; i ::; 3. (3.9)
We denote by w;, w~, w:i the 1-forms associated to this frame. Since dX' = w; e; + w~e.~, it follow::. from (3.8) that
From (3.9) we have 1 ::; i,j ::; 3.
Now
(H- aK)w11\. w2 = H'(l- 2aH + a2 K)wtl\. w2
hence we proved the first relation of (3.6). Similarly from equation
we obtain
W13 1\ W23 = K 1(1 - 2aH + a2 l{)w1 1\ w2
and hence the first equation of (3.7).
(3.10)
(3.11)
II) Now we consider parallel surfaces in the unit sphere M = S3 c Jti. We will consider a local parametrization X : U C R2 -+ S3 C R:' for M and an orthonormal frame el> e2, e3 adapted to M, where f3 is normal to M and tangent to 83 . Then the parallel surface is locally given by
X' = cos aX + sin ae3 •
Hence (3.12)
where cos aw1 - sin aw13 and cos aw2 - sin aw23 are linearly independet as a con sequence of (3.4). We consider the frame adapted to X', given by
(3.13)
From (3.12) and (3.13) the 1-forms associated to this frame satisfy the following relations
w;3 = sin aw1 + cos aw13, w~3 = sin awz + cos awz3·
From these relations and equation (1.11) for the surface X', it follows that H' is given by the second expression of (3.6). Similarly, from (1.9) we obtain the second equality of (3. 7).
III) If M is a surface in H3 C L\ we will consider a local parametrization X : U C R2 ~ H3 C L\ where L" is the Lorentzian four-dimensional space and lXI = -1. Let e11 e2, e3 be an orthonormal frame adapted to M in H3. Then the parallel surface is locally given by
X'= cosh aX+ sinhae3 •
Hence dX' = (cosh awz -sinh aw,a)et + (cosh aw2 -sinh aw23)e2
where cosh aw1 -sinh aw13 and cosh aw2 - sinh aw23 are linearly independent as a consequence of (3.4). We consider the frame adapted to X' C H3 given by
e~ = e" e~ = ez, e; = sinh aX + cosh ae3.
Then, the 1-forms for this frame satisfy the following relations:
w; = cosh aw1 - sinh aw13, w~ = cosh aw2 - sinh aw23,
tv; 2 = W12 1 w;3 = sinh aw1 cosh aw13, w~3 = sinh awz + cos haw23
Using (1.11) and (1.9) forM' and M, we get the third equalities of (3.6} and (3.7). 0
3.3 Remark. It follows from the proof of Proposition 3.2 that parallel surfaces have the same principal directions.
As a consequence of the previous proposition we obtain our next result, which is known as Bonnet's theorem when the ambient space is Euclidean.
3. Geodesic congruences and parallel surfaces 27
We recall that a point p of a surface M contained in M 3 (K) is said to be umbilic if the two principal curvatures at p are equal. It is easy to see that p is umbilic if and only if (H2 - (K- K)](p) = 0, where Hand f{ are the mean and Gaussian curvature of M.
3.4 Corollary. Let M 2 be a regular orientable surface in M 3(K) with no umbilic points and such that its Gaussian curvature does not vanish. If M has constant
mean curvature H = c > 0 ( c > 1 if K = -1) then there exist two surfaces in M parallel to M such that one has constant positive Gaussian curvature
K = 2 (c2 + ]( +cVc2 + K) (3.14)
and the other one has constant mean curvature equal to -c.
ContJersely, for each surface M in M3(J<) with no umbilic points whose Gaussian curvature is a positive constant K, K > 2 if K = 1, there exist two surfaces parallel to M whose mean curvatures are constant equal to c and -c respectively,
where c = IK- 2KI/(2V K- K.
Proof. Let X : U C R2 --+ M3(J<) be a local parametrization of the surface
M C M 3 • Assume M has constant mean curvature H = c > 0 ( c > 1 if K = -1 ). We consider a parallel surface defined by
{ X + ae3 if I< = 0,
X'= cos aX+ sinae3 if K = 1, cosh aX +sinhae3 if K = -1,
(3.15)
where e3 is a unit vector field normal to M for which H = c and
a = l/(2c) if K=O,
tan 2a = 1/c if /{ = 1, (3.16)
tanh 2a = 1/c if K = -1.
Since, K does not vanish, it follows from equation (3. 7) that the Gaussian curvature K' of X' is given by (3.14), where we use the identities
• 2 1 1 SID a = - - -::-r====;;:===
2 2v'1 + tan2 2a
Similarly, if we consider X' as in (3.15) where
a= 1/c if K = 0, tan a= 1/c if K = 1,
tanh a= 1/c if K = -1,
(3.17)
we conclude that X' has constant mean curvature -c. This follows from (3.6) and the fact that M has no umbilic points. From (3.14), we observe that c2 + K > 0 and c > 0 imply K > 0. In particular, K > 2 when K = 1.
Conversely, assume X is a local parametrization for a surface M C M 3 (I<), whose Gaussian curvature is a positive constant K, K > 2 if K = 1. We consider the two parallel surfaces
X ± ae3 if K = 0, cos aX± sin ae3 if I<= 1,
cosh aX ± sinh ae3 if K = -1, (3.18)
where e3 is a unit vector field normal toM, a is defined as in (3.16) and cis given
by c = \K- 21<\/(2/I<- I<. Observe that/(> 0 implies that c > 1 if K = -1. As a consequence of Proposition 3.2, the identities (3.17) and the fact that M
has no umbilic points, we obtain that the parallel surfaces defined by (3.18) have constant mean curvature =fc respectively.
0
3.5 Example. We consider the surface of Delaunay X(x 11 x2 ) given in Example 2.9, whose mean curvature is H = 1. It follows from Corollary 3.4 that the surface X', parallel to X at a distance equal to 1/2, has constant Gaussian curvature K' = 4 and the surface X" parallel to X at a distance equal to I has constant mean curvature H" = -1.
As a consequence of Proposition 3.2 we have the following results where without loss of generality we assume I< = 0, 1 or -1.
3.6 Corollary. Let M and M' be two parallel surfaces in M 3 (I<). M is a linear Weingarten surface if and only if M' is a linear Weingarten surface. In particular, M is a fiat surface if and only if M' is a fiat surface.
Proof. Let M and M' be parallel surfaces of M at a distance a. Then the Gaussian and mean curvatures I< and H of M are related to the curvatures I<' and H' of M' by the relations (3.6) and (3. 7).
It follows that H and [{ satisfy a linear equation of the form
a + 2(3 H + 1U< - I<) = 0
if and only if
a+ 2((3 + o:a)H' +(I+ 2(3a + aa2 )f(' = 0 if K=O;
o:- (o: -1) sin2 a+ ,8 sin 2a + [2,8 cos 2a + (o: -1) sin 2a]H' + +[a- (a -1) cos2 a+ ,8sin2a](K' -1) = 0 if 1? = 1;
3. Geodesic congruences am/ parallel surfaces 29
a+ (a+ "f)sinh2 a+ /3sinh 2a + [2/3cosh 2a +(a+"') sinh 2a]H' + +[-a+ (a+ "f) cosh2 a+ .Osinh 2a](K' + 1) = 0 if [( = -1;
In particular, it follows from (3.7) that [{ = 0 if and only if K' = 0. 0
3. 7 Corollary. Let M be a linear Weingarten surface in M 3(K) satisfying
a+ 2/3H + "f(l<- K) = 0
where "{/3 =f. 0. {i} Suppose a-"{ [( = 0 and assume also bl < l/31 if [{ = 1. Then given a generic point p0 of M thtre exists a neighborhood of Po which is parallel to a minimal
surface. {ii} Suppose a- "{K =f. 0 and assume also 12/31 < !a+ "'I if K = -1. Then given a generic point Po of M there exists a neighborhood of Po which is parallel to a surface of constant Gaussian curvature.
Proof. If a-"' K = 0 we consider a parallel surface M' at a distance a defined by
a = -"Y/(2/3) sin 2a = -"{ //3 sinh2a = -"{//3
if [( = 0, if [( = 1, if [( = -1.
The surface M' is defined on a neighborhood of a generic point where (3.5) holds. Then, it follows from Proposition 3.2 that H' = 0, i.e. M' is a minimal surface. If a - "'K =f. 0 taking the distance a to be
a=-/3/a if K=O, tan2a = 2(Jj(a- "') if K = 1,
tanh 2a = 213/(a + /) if K = -1,
we get a parallel surface M' defined on a neighborhood of a generic point. It follows from Proposition 3.2 that M' has constant Gaussian curvature.
0
3.8 Examples. a) The family of linear Weingarten surfaces described by Example 2.6 consists
of surfaces parallel to the pseudo-sphere at a distance a (see Fig. 2.3). Each surface satisfies the relation 1 + 2aH + (1 + a2 )K = 0.
b) The surfaces given in Example 2.11 provide a 1-parameter family of Wein garten surfaces parallel to the Enneper surface at a distance a (see Fig. 2.4). For each surface we have H - aK = 0.
c) Consider a Clifford torus
X(x1 , x2 ) = (c1 cos X1, Ct sinx1,c2 cos x2 , c, sin x2 )
where c1, c2 are nonzero real constants such that c~ + c~ = 1. Then a surface X' parallel to X in f:P at a distance a is also a Clifford torus, where the constants are given by c'1 == c1 cos a+ c2 sin a and c; = c2 cos a- c1 sin a.
§4. Pseudo-spherical geodesic congruences
In this section, we shall study pseudo-spherical geodesic congruences in a space form of constant curvature K. Pseudo-spherical line congruences were originally studied by A.V. Backlund [Ba1,Ba2: in 1875, for surfaces in the Euclidean three space. He considered two diffeomorphic surfaces Af and M' in R3 such that cor responding points p and p' determine a line which is tangent to both surfaces and has constant length independent of p. He also assumed that the angle between the normal vectors at p and p' is constant independent of p. Then Backlund proved that a pseudo-spherical line congruence exists only between surfaces whose Gaus sian curvatures are both equal to the same negative constant (this justifies the denomination of pseudo-spherical co:1gruence). Moreover, he showed that given a surface of constant negative curvature one obtains a two-parameter family of sur faces of the same curvature. Hence he obtained a transformation between surfaces of constant negative curvature, originating what is now called a Backlund trans formation. Later, Bianchi [Bi3] studied compositions of Backlund transformations obtaining the Permutability theorem. He also considered similar congruences in
space forms M 3 (K) [Bil,Bi2]. The goal of this section is to prove these results for surfaces in space forms
lvP(K), where without loss of generality we assume K = 0, 1 or -1. We will show that a pseudo-spherical geodes:c congruence exists only between surfaces in
M 3 ( K) whose Gaussian curvatures are both equal to the same constant, K < [(. Moreover, we will prove that given such a surface M 2(K) C M 3(K), there exist a two-parameter family of pseudo-spherical geodesic congruences between M and a family of surfaces M'. This will provide a transformat:on between surfaces with the same constant curvature K < K. The composition of such transformations will be considered in the Permutability theorem.
4.1 Definition: Let f : M ~ M' be a diffeomorphisrr: between two surfaces in
M 3(K) such tha.t for each p E M, and p::::: f(p) :f: p there exists a unique geodesic
in M joining p and p' whose tangent vectors at p and p' are in TpM and Tp•M' respectively. We say that f. is a pseudo-spherical geodesic congruence if
1) the distance between p and p' on M is a constant r, independent of p;
2) the angle between the normals Np and N;, is a constant 0, independent of p;
The definition above is an extension of the notion of pseudo-spherical line congru ence in R3 introduced by Backlund in his classical work [Bal].
4. Pseudo-spherical geodesic congruences 31
4.2 Backlund's theorem in space forms. Let M and M' be two surfaces con
tained in M 3 ( I<). Assume f. : M -4 M' is a pseudo-spherical geodesic congruence such that the distance between corresponding points p, p' is a constant r > 0 and the angle between the normals at p and p' is a constant 8, 0 < 8 < 1r. Then, both M and M' have constant Gaussian curvature [(, where
sin2 () if [{ = 0, ---r2
if [(;; 1, ( 4.1)
if [{ = -1.
Proof. i) Let M and M' be surfaces of the euclidean space R3 . Consider M and M' locally given by parametrizations X(x 1 , x2 ) and X'(xt, xz) with (x 1, x2) E U C R2, where U is an open subset of R2• Let ell e 2 , e 3 be an orthonormal frame adapted to M, such that e1 is in the direction of the line congruence, i.e.
(4.2)
Let w1, w2, Wij be the dual and connection forms associated to this frame. Since f. is a pseudo-spherical geodesic congruence, there exists a local frame for M' defined by
e~ = cos 8e2 +sin Oe3 , ( 4.3)
e; = -sin Oez +cos Oe3.
We denote by w;, w;, w:i the 1-forms associated to this frame. Differentiating ( 4.2) and using (1.1) we obtain
dX' = WJft + (wz + rW12)e2 + rw13e3.
On the other hand,
dX' = w~ e~ + w~e~ = -w; e1 + w;( cos Oe2 +sin Oe3 ).
Therefore, comparing these expressions, we get
w~ = -wt,
w2 sin 0 rwl3·
1 w12 = - -w2 + cot Ow13.
r
( 4.4)
(4.5)
Taking the differential of w1 2, it follows from the structure equations (1.7) and (1.9) that
1 dw12 ::::: --dw2 +cot Odw1a
r 1
== w12/\ (~wt +cot fJw2a).
Substituting w12 by its expression given by (4.5) and using Gauss equation (1.7) and (1.9), we obtain
We conclude using again equation (1.9) that
• 2 () [{ = _ sm
2 •
r
By symmetry, we conclude that M' has the same constant Gaussian curvature.
ii) Let M and M' be surfaces of the unit sphere S3 contained in the Euclidean space R". Consider M locally parametrized by X : U c R2 --+ M c S3 c R". Since there is a pseudo-spherical congruence between M and M', there exist orthonormal frames e., e2 , e3 for M and e~, e~.e; for M' tangent to S3 such that
e~ = sinrX- cos reb
e; == cos 8e2 + sin Oe3 ,
e; = -sin Oez + cos Oe3 ,
where at each point p E M, e1 is the unit vector tangent to the geodesic joining p top' = i(p) and e3 (resp. e;) is normal to M (resp. M') at p (resp. p').
Let X' denote the position vector forM', then
X'== cosrX +sinre1•
We observe that dX = w1e1 + Wze2, therefore we have (dei>X) = -w1 and (de2, X) = -w2. Hence,
On the other hand
dX' = w~e~ + w;e; = w~(sin rX- cos re1) + w~(cos0e2 + sin0e3 ).
4. Pseudo-spherical geodesic congruences
w' 1 = -w1,
sin Ow~ = sin rw13.
w12 = - cot rw2 + cot Owt3·
Taking the differential of w12 , we obtain
dw12 = -cot rw1 1\ Wtz + cot Ow12 1\ W23
= w12 1\ (cot rw1 +cot Owz3).
33
(4.6)
(4.7)
Hence, using (4.7) and the structure equations (1.7) and (1.9) where I<= 1, we get
dw12 = ( cot2 r + cot2 O(I<- 1 ))wt 1\ w2.
On the other hand, since dw12 = -I< Wt A w2, we conclude that
By symmetry M' has the same Gaussian curvature.
iii) If M and M' are surfaces of H3 , we consider H3 contained in a Lorentzian space L'' a.nd M locally parametrized by X : U C R2 -t M C H3 C L 4, where U is an open subset of R2. Then the position vector X is normal to H3 and IIXII = -1. We consider orthonormal frames e1, e2, e3 for M and e~, e~, e; for M' tangent to H3 such that
e~ = sinh r X -cosh re1,
e~ = cos Oe2 +sin Oe3,
e; = - sen0e2 + cos Oe3 ,
where e1 is tangent to the geodesic of the congruence l and e3, e; are normal to the surfaces M and M' at corresponding points. Then M' is locally given by
X'= cosh r X- sinh re1,
with arguments similar to the previous case we get
w~ = -w~,
sin8w~ = -sinhrw13 ,
We conclude that M and M' have constant curvature
I< + 1 = _ sen21J . sinh2 r
(4.8)
0
4.3 Remark. Backlund's theorem can be extended to a transformation between linear Weingarten surfaces. Consider a diffeomeorphism £ : M -t M' between surfaces M and M' contajned in M 3(K) such that for corresponding points p and
p' = l(p) there exists a unique geodesic in M joining p top' whose tangent vectors at p and p' form constant angles with the normal vectors Np and NP' and conditions 1) and 2) of Definition 4.1 hold. Then one can prove that M and M' are linear Weingarten surfaces {see [Bi4]).
4.4 Remark. It is not difficult to prove that a pseudo-spherical geodesic congru ence preserves lines of curvature and asymptotic lines.
The proof of Theorem 4.2 shows that for a pseudo-spherical geodesic congruence to exist it is necessary that equations ( 4.5), ( 4. 7) and ( 4.8) hold for K = 0, 1 and -1
respectively. Our next result shows that given a surface M C M 3(K) of constant Gaussian curvature K satisfying (4.1) there exists a 2-parameter family of surfaces M' whkh are related locally to M by pseudo-spherical geodesic congruences. The surfaces M' are obtained by integrating (4.5), {4.7) and {4.8) for K = 0, 1 and -1 respectively. This equation is called Backlund transformation and it will be denoted by BT(O), where(} is the parameter.
4.5 Integrability Theorem. Let M 2(K) c M 3(K) be a surface of constant curvature /{ < K satisfying (4 .1} where r > 0( r < 1r if K = 1) and 0 < () < 1r
are constants. Given a point Po E M and Vo E T Po M a unit vector which is not a
principal direction, there exists a surface M' of M and a pseudo-shperical geodesic congruence f. between a neighborhood of p0 and M', such that the geodesic joining
Po to P6 = l(p0 ) is tangent to v0 at po, the distance in M between Po and p~ is r and IJ is the angle between the normals at Po and p~.
Proof. We will start showing that there exists a frame e1 , e2, e3 , adapted to M, defined on a neighborhood of Po, such that the relation
{ -~w2 +cot Owta if K = 0,
W12 = -cot rw2 +cot 8w13 if K = 1,
coth rw2 + cot 8w13 if K = -1
(4.9)
4. Pseudu-~:~pherical g·eodesic congruences
holds and e1 (Po) = vo Let I be the ideal generated by the 1-form
w12 + -w2 - cot Ow13 {
Wt2 - coth rw2 - cot Ow13
35
if /{ = -1.
{
1 -w1 + cot Ow23
d7 = - K Wt /1. W2 - Wt2 A ;ot ru:1 + cot Ow23
- coth rw1 + cot Ow23
- (-.-1-J( + _.!.._) Wt /1. W2 sm2 (} r 2
if /{ = 0,
d,=. - [-.-1-(K -1) + -.-1-] Wt /1. w2 sm20 sm2 r
if /{ = 1, (mod I)
-[ .\ 0(K+l)++]wtAW2 if K= -1. sm sm r
By hypothesis on the curvature K, we get d7 =. 0 (mod I). It follows from Frobenius Theorem that there exists a frame defined on a neighborhood V of Po such that e1{p0 ) = v0 and the Backlund tranformation (4.9) is satisfied. Since Vo is not a principal direction at p0 we may assume that V is sufficiently small so that e1 is not a principal direction at each point of V. Moreover, we may consider V parametrized by X: U c R2 -+ V c M c M3 (K), where M 3(K) = S3 C R4 for
"K = 1 and M 3(K) = H3, when/(= -1, is contained in Lorentzian space L4 • We define X' by r+re, if [{ = 0,
X'= cosrX + sinre1 if /( = 1, (4.11)
cosh X - sinh re1 if K= -1.
We need to prove that M' = X'(U) is an immersed surface in J,f3 and f : V -t M 1
is a pseudo-spherical geodesic congruence. From {4.11) we get
{ w1e1 + (w2 + rw12)e2 + rw13e3
dX' = cos r( w1 e1 + wzez) + sin r{ w12e2 + w13e3 - w1X)
cosh r( w1 e1 + w2e2) - sinh r( w12e2 + w13e3 + w1X)
if K=O,
if K= 1,
if K= -1.
Therefore, using the Backlund transformation ( 4.9) to substitute w12 , we obtain
dX'=
. sinr Wt(cos re1- sm rX) + ~Wta(cos8ez + sin8e3) if [{ = 1,
Slfi'l
sm
Since e1 is not a principal direction in V, it follows that w1 and w13 are linearly independent. Therefore, X' defines a surface of M3 and the vector field cos He2 + sin Oe3 is tangent to X'. Moreover, we conclude that e1 is a tangent direction to both surfaces and e~ = - sen8e2 + cos 8e3 is normal to X', hence () is the angle between e3 and e~. From the definition of X', given by ( 4.11) the distance between corresponding points of X and X' is equal to the constant r.
0
The integrability theorem above provides a method of obtaining, from a given surface of constant curvature M(I<) C JVP(I<) with I<</(, a two-parameter fa mily of such surfaces, by integrating the Backlund transformation. The parameters are () and the one that corresponds to choosing the unit vector v. Observe that r is determined by K and 0 from the relation ( 4.1).
4.6 Remark. Let M(K) be a given surface of constant curvature contained in
M 3(K) where I<< K. Assume M is parametrized by X(xt,x2) as in Corollary 2.7. Then
- X:rz ez=- sinf
2
are the principal directions of M where 1/J satisfies the sine-Gordon equation. The 1-forms associated to this frame are given by
- 1/Jd W1 =COS 2 Xt
and
w2a = VIK- Kf cos ~dx2. Let rand() be constants such that (4.1) is satisfied. We fix a unit vector v0(x~, x~) tangent to M at X(x~, x~), which is not a principal direction. It follows from the integrability theorem that there exists a moving frame e1 , e2, e3 adapted to M, such that e1(xy,xg) = v0 and (4.9) is satisfied for the associated 1-forms whw2,w12.
4. Pseudo-spherical geodesic congruences 37
Hence, in order to determine the vector field e1, we look for a function 1/;' such that
1/J' ~ . 1/J' ~ e1 = cos 2 et + sm 2e2
• 1/J' - + 1/J' - e2 = -sm 2e1 cos 2e2.
and the associated forms, given by
1/J' ~ . 1/J' - Wt cos 2Wt + sm 2w2,
. 1/J' - t/J' - W2 = -SID 2to1 +COS 2W21
1/J' - Wt2 = d2 + w12,
satisfy ( 4.9), i.e.
where
1/J~ 1 + tP:~:2 = 2sin f cos t.e(r) + 2 cot 9 cos f sin t cot 9/IK- KJ,
t/J~2 + 1/lz1 = -2cos ~sin ~l(r)- 2cot Osin fcos ~cot 9y'!K- Kl,
{ 1/r
if K=O, if K= 1, if K = -1.
Using (4.1), and normalizing IK- Kl = 1, this system reduces to
where
/. I 2 • !fL tb ( il) ll t/J' • !£. n.,1 + '/'"'• = RJn 3 COR '2 [, u + 2 cot u COS T Sin 2
tP~2 + 1/lz1 = -2cos f sin ~L(O)- 2cot 8 sin f cos~
if K = 0, if K = 1, if K = -1.
We conclude from the Integrability theorem that
(=f sin£ ) if K=O, X+r X:~: 1 +~X.,2 cos 2 sln
X'= cosrX + sinr ( :;:fx.,. + :fx.,2) if K=l,
. (=i ein£ ) coshrX -smhr X,1 +~X%2 COS 2 IDD if K=-1.
(4.12)
(4.13)
is a new surface of constant curvature /(in M 3 (I<). We conclude this remark by observing that ( 4.9) is equivalent to ( 4.12) and
moreover, (4.13) provides explicitly the surface X' associated to X by the Backlund transformation.
4. 7 Example. We consider the pseudo-sphere in R3 parametrized by
We have seen in Example 2.2 that we associate to X a function '1/J defined by
1/-' cos 2 = tanh x1 . 1/J h sm 2 =sec x 1•
which is a solution of the differential equation
In order to obtain a new surface of constant negative curvature associated to X by the Backlund transformation, we need to solve (4.12). In particular, if we choose() = 1rj2, equation (4.12) reduces to
?jl 'I/J~ 1 = 2 sin 2 tanh x 11
1/J' 1/J~2 = 2( 1 -cos 2 )sech Xt.
The solutions of this system are given by
'1/J' cot 4 = ( -x2 + c)sech x 1•
Fig. 4.1 Kuen's surface
Hence, we get Kuen's surface, which is a constant negative curvature surface ob tained from the pseudo-sphere by a Backlund transformation BT( 1r /2), given ex plicitly by ( 4.13) , i.e.
1/J' '1/J' X'= X+ cos 2cothx1X,1 +sin 2coshx1Xr2 •
Fig. 4.1 shows Kuen's surface, given by X'(xh x2 ), when we consider the domain (x 1.x2 ) E [-4,4] x [-4.45,4.45]. If we extend the domain to (xl!x2 ) E [-4,4] x [ -6, 6], we obtain the surface shown in Fig. 4.2 from the side, front and back views respectively.
4. Pseudo-spherical geodesic cong1·ueiJces
Fig. 4.2 Kuen's surface of constant negative curvature obtained from the pseudosphere by a Backlund transformation BT( 1r /2).
39
Our next result shows that the composition of Backlund transformations is commutative, i.e. the following diagram holds
M' BT(9t) BT(02)
M"
4.8 Permutability Theorem. Let M, M', M" 6e surfaces immersed in a space
form M 3(K). Suppose there exist pseudo-spherical geodesic congruences f 1 : M-+ M' and e2 : M -+ M" with constants rl! 01 and r2, 02 respectively, Ot =I= 02. Then
there exists a surface M* C M and pseudo-spherical congruences f2 : M' -+ M•, li : M" -+ M* with constants r 2 , 02 and Tt. 01 respectively, such that
Proof. Let v11 v2 be a local frame given by principal directions on M. Consider frames e11 e2 and e11 e2 for M,e~,e~,e~ forM' and e~,e;,e3 for M 11 as in Theorem 4.2, i.e.
if K:: 0; if K:: 1; if K:: -1;
40 I. Transformations of surfaces a.nd applications
if K:: 0; if K:: 1; if K:: -1;
e~ = cos 9t e2 + sin 61 ea e~ = cos 62e2 + sin 02ea
ea = -sin 81 e2 + cos 01 ea e~ = -sin 02e2 + cos 02ea,
(4.14)
where X denotes the position vector for M and e1 , e1 are the unit vectors, tangent to the geodesic from p top'= it (p) and p" = i 2 (p) respectively.
We denote by
if K:: 0; if /( = 1; if K = -1;
if K :0; if [( = 1; if K::: -1;
the position vectors of M' and M" respectively. We consider the following matrix notation
Moreover, we denote by Di the matrix defined by
where
{ 1/r
if [( = 0; if K:: 1; if [( = -1;
for the constants Ti,(Ji, where i = 1,2. Let C be the orthogonal matrix defined by
1~i,j~2.
where
( 4.15)
(4.16)
(4.17)
(4.18)
{4.19)
We remark that since
(4.20)
it follows that E 1 E = F' F, hence H is an orthogonal matrix. We define tangent frames e~ on M' and e~' on M" by
( 4.21)
We consider the pseudo-spherical geodesic congruences defined by the ma.ps
if K::O; if K::: 1; if K::: -1;
if K :::0; if [{ = 1; if K::: -1;
(4.22)
(4.23)
In order to complete the proof of the theorem we only need to prove that t; o£1 = £~ o£2 . In fa.ct, :1sing ( 4.14), ( 4.16), ( 4.18) and ( 4.21 ), we obtain e; o f1 (X) and £i o l 2(X). The equality follows from (4.19) and Theorem 4.2.
0
4.9 Example. We will illustrate the Permutability Theorem in a composition of Backlund transformations applied to the pseudo-sphere. In Example 4. 7 we considered the pseudo-sphere (see Fig. 4.3) given by
which is associated to the solution of the sine-Gordon equation '1/J given by tan( t/J I 4) = ez•. By applying the Integrability Theorem, we solved the Backlund transforma tion BT(O), where(} = 1r12 and we obtained a new surface of constant negative curvature (see Fig. 4.4) given by
tP2 . tP2 x2 = X + cos 2 coth xlx.,, + SID 2 cosh XtX.,,,
where ,p· cot 4
2 = ( -x2 + c)sech x 1.
Similarly, if we consider I) = '11' I 4 we obtain another surface associated to X by BT(-:r/4) (see Fig. 4.5), given by
J2 ( tPl h . tPl ) xl = X + 2 cos 2 cot ZlX.,, + SID 2 cosh x1X.,2 ,
where t/J1 r.:: e v'2z,-z2 _ e:r:'
tan - = ( v 2 + 1) . 4 1 + ev'2:r:,-:r:2+z1
It follows from the proof of the Permutability Theorem that the matrix C defined by (4.18) is given by
c- ( cos(t/Jt-'I/J2) sin(tbt-'I/J2)) - -sin(tbt- tb2) cos(tbt- tP2) ·
Fig. 4.4
Fig. 4.5
Fig.4.3 is the pseudo-sphere. Fig.4.4 and Fig.4.5 are surfaces obtained from the pseudo-sphere by the Biicklund transformations BT( 1r /2) and BT( 1r /4) respectively. Fig.4.6 is a surface obtained from Fig.4.4 and Fig.4.5 by the transformations BT(7r/4) and BT(7r/2) respectively.
5. Biickluml tnt..u::;[onnation. Superposition formula 43
By computing the matrix B defined by ( 4.19), we conclude that the fourth surface x· (see Fig. 4.6), which is associated to X1 by BT(1rj2) and to X2 by BT( 1r /4) is given explicitly by
X = X2 +- -Buet + -B12e3 . V'i( ../2 ) 2 2
where e3 is the normal vector to X, e1 = cos(t/J2/2)e1 + sin(t/J2/2)e2 , and
e2 = Xx-z/sin(t/J/2).
The analytic interpretation of the geometric construction given by the pseudo spherical geodesic congruences will be treated in the following section in terms of solutions of the sine-Gordon equation. A generalization of Backlund's theorem and of the Permutability theorem to higher dimensional manifolds will be given in §1 and §2 of Chapter III.
§5. Backlund transformation for the sine-Gordon and the elliptic sinh-Gordon equations. Superposition formula
In section 2 we have seen that classes of linear Weingarten surfaces are associated to solutions of the sine-Gordon equation 1/.Jx1x1 - 'I/Jx2 :c2 = sin'¢, or to the elliptic sinh-Gordon equation tPx1x1 + tPx2r2 = -sinh t/J. In particular, as we have seen
in Corollary 2.7, surfaces of nonzero constant curvature M(K) C M 3(K) where K < K were shown to correspond to solutions of the sine-Gordon equation and surfaces for which K > K were associated to solutions of the elliptic sinh-Gordon equation. Both equations are highly nonlinear and they are not easy to solve. In this section, we will describe a method of obtaining solutions for these equations by providing what is known as a Backlund transformation and a superposition formula. for both equations.
A Backlund transformation (sometimes c

transformations of manifolds and application to differential equations

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