analytic and diﬀerential geometry mikhailg - biuu.math.biu.ac.il/~katzmik/egreglong.pdf ·...

Analytic and Differential geometry

Mikhail G. Katz∗

Department of Mathematics, Bar Ilan University, Ra-

mat Gan 52900 Israel

E-mail address : [email protected]

∗Supported by the Israel Science Foundation (grants no. 620/00-10.0and 84/03).

Abstract. We start with analytic geometry and the theory ofconic sections. Then we treat the classical topics in differentialgeometry such as the geodesic equation and Gaussian curvature.Then we prove Gauss’s theorema egregium and introduce the ab-stract viewpoint of modern differential geometry.

Contents

Chapter 1. Analytic geometry 111.1. Course notes 111.2. Linear algebra, index notation 131.3. Einstein summation convention 141.4. Quadratic forms, polarisation formula 141.5. Matrix as a linear map 151.6. Symmetrisation and skew-symmetrisation 161.7. Matrix multiplication in index notation 161.8. Two types of indices: summation index and free index 171.9. The inverse matrix, Kronecker delta 181.10. Eigenvalues, symmetry 191.11. Euclidean inner product 20

Chapter 2. Eigenvalues of symmetric matrices, conic sections 212.1. Finding an eigenvector of a symmetric matrix 212.2. Trace of product in index notation 222.3. Inner product spaces and self-adjoint operators 232.4. Orthogonal diagonalisation of symmetric matrices 232.5. Classification of conic sections: diagonalisation 252.6. Classification of conics: trichotomy, nondegeneracy 272.7. Characterisation of parabolas 282.8. Summary: classification of conics 29

Chapter 3. Quadric surfaces, Hessian, representation of curves 313.1. Quadric surfaces 313.2. Case of eigenvalues (+ + +) or (−−−), ellipsoid 313.3. Determination of type of surface 323.4. Case of eigenvalues (+ +−) or (+−−), hyperboloid 333.5. Rank 2, hyperboloid, cylinder 343.6. Exercise on index notation 363.7. Hessian, minima, maxima, saddle points 373.8. Parametric representation of a curve 383.9. Implicit representation of a curve 393.10. Implicit function theorem 39

3

4 CONTENTS

3.11. Unit speed parametrisation 40

Chapter 4. Curvature, Laplacian, Bateman–Reiss 414.1. Geodesic curvature 414.2. Gauss map; shape operator for curves 424.3. Osculating circle of a curve 444.4. Center of curvature, radius of curvature 454.5. Flat Laplacian 454.6. Bateman–Reiss operator 464.7. Geodesic curvature for an implicit curve 464.8. Curvature of conic sections 474.9. Maximal curvature of a transcendental curve 484.10. Exploiting the defining equation in studying curvature 494.11. Curvature of graph of function 50

Chapter 5. Surfaces and their curvature 515.1. Existence of arclength 515.2. Non-existence of regular parametrisation 525.3. Arnold’s observation on folding a page 525.4. Regular surface; Jacobian 535.5. Coefficients of first fundamental form of a surface 545.6. Metric coefficients in spherical coordinates 545.7. Tangent plane 555.8. First fundamental form of M 555.9. Ip of plane and cylinder 565.10. Surfaces of revolution 575.11. From tractrix to pseudosphere 585.12. Chain rule in two variables 59

Chapter 6. Γ symbols 616.1. Measuring length of curves on surfaces 616.2. Normal vector to a surface 626.3. Γ symbols of a surface 636.4. Basic properties of the Γ symbols 636.5. Derivatives of the metric coefficients 646.6. Intrinsic nature of the Γ symbols 656.7. Γ Symbols for a surface of revolution 666.8. Spherical coordinates 676.9. Great circles 686.10. Position vector orthogonal to tangent vector of S2 69

Chapter 7. Clairaut’s relation, geodesic equation 717.1. Clairaut’s relation 71

CONTENTS 5

7.2. Spherical sine law 717.3. Longitudes 737.4. Proof of Clairaut’s relation 747.5. Differential equation of great circle 757.6. Metrics conformal to the flat metric and their Γkij 777.7. Gravitation, ants, and geodesics on a surface 787.8. Geodesic equation 797.9. Equivalence of definitions 80

Chapter 8. Geodesics on surface of revolution, Weingarten map 818.1. Geodesics on a surface of revolution 818.2. Integration in polar coordinates 838.3. Integration in cylindrical coordinates 838.4. Integration in spherical coordinates 848.5. Measuring area on surfaces 858.6. Directional derivative 868.7. Extending v.f. along surface to an open set in R3 878.8. Differentiating normal vectors n and N 888.9. Hessian of a function at a critical point 888.10. Definition of the Weingarten map 898.11. Properties of Weingarten map 90

Chapter 9. Weingarten map, Gaussian curv., second fund. form 919.1. Weingarten map is self-adjoint 919.2. Relation to the Hessian of a function 929.3. Weingarten map of sphere 939.4. Weingarten map of cylinder 939.5. Coefficients Lij of Weingarten map 949.6. Gaussian curvature 949.7. Second fundamental form 959.8. Geodesics and second fundamental form 969.9. Three formulas for Gaussian curvature 97

Chapter 10. Principal curvatures, minimal surf., thm. egregium 10110.1. Principal curvatures of surfaces 10110.2. Mean curvature, minimal surfaces 10210.3. Minimal surfaces in isothermal coordinates 10410.4. Minimality and harmonic functions 10610.5. Introduction to theorema egregium; intrinsic vs extrinsic 10710.6. Riemann’s formula 10810.7. Review of notation 10910.8. An identity involving the Γs and the Ls 11010.9. The theorema egregium of Gauss 111

6 CONTENTS

Chapter 11. Signed curvature of plane curves 11311.1. Curvature expressed in terms of θ(s) 11311.2. Signed curvature with respect to an arb. parameter 11411.3. Jordan curves in the plane 11511.4. Gauss map 11511.5. Convex Jordan curves 11611.6. Total curvature of a convex Jordan curve 11811.7. Signed curvature kα of Jordan curves in the plane 11911.8. Rotation index of a closed curve in the plane 12011.9. Total signed curvature 120

Chapter 12. Surfaces, hyperbolic metric, Gauss–Bonnet 12312.1. Gaussian curvature as product of signed curvatures 12312.2. Mean versus Gaussian curvature 12412.3. The Laplace–Beltrami operator 12412.4. Laplace-Beltrami formula for Gaussian curvature 12512.5. Hyperbolic metric in the upperhalf plane 12612.6. Area elements of the surface, orientability 12812.7. Gauss map for surfaces in R3 12912.8. Sphere S2 parametrized by Gauss map of M 12912.9. Comparison of two parametrisations 130

Chapter 13. Gauss–Bonnet theorem 13313.1. Special case of Gauss–Bonnet 13313.2. Proof of Gauss–Bonnet Theorem 13413.3. Local Gauss–Bonnet 13513.4. Euler characteristic 13513.5. Rotation index of closed curves and total curvature 13613.6. Surfaces of revolution and isothermalisation 13613.7. Curvature of pseudosphere 138

Chapter 14. Duality in linear algebra, calculus, diff. geometry 14114.1. Transition from classical to modern diff geom 14114.2. Duality in linear algebra; 1-forms 14114.3. Dual vector space 14214.4. Derivations 14314.5. Characterisation of derivations 14414.6. Tangent space and cotangent space 14514.7. Constructing bilinear forms out of 1-forms 14614.8. First fundamental form 14614.9. Dual bases in differential geometry 14714.10. More on dual bases 148

CONTENTS 7

Chapter 15. Lattices and tori 15115.1. Circle via the exponential map 15115.2. Lattice, fundamental domain 15215.3. Fundamental domain 15215.4. Lattices in the plane 15315.5. Successive minima of a lattice 15415.6. Gram matrix 15515.7. Sphere and torus as topological surfaces 15515.8. Standard fundamental domain 15615.9. Conformal parameter τ of a lattice 15715.10. Conformal parameter τ of tori of revolution 15915.11. θ-loops and φ-loops on tori of revolution 16015.12. Tori generated by round circles 16015.13. Conformal parameter of tori of revolution, residues 161

Chapter 16. A hyperreal view 16316.1. Successive extensions N, Z, Q, R, ∗R 16316.2. Motivating discussion for infinitesimals 16416.3. Introduction to the transfer principle 16516.4. Transfer principle 16816.5. Orders of magnitude 16916.6. Standard part principle 17116.7. Differentiation 171

Chapter 17. Global and systolic geometry 17517.1. Definition of systole 17517.2. Loewner’s torus inequality 17917.3. Loewner’s inequality with defect 18017.4. Computational formula for the variance 18117.5. An application of the computational formula 18117.6. Conformal invariant σ 18217.7. Fundamental domain and Loewner’s torus inequality 18217.8. Boundary case of equality 18317.9. Global geometry of surfaces 18417.10. Definition of manifold 18517.11. Sphere as a manifold 18617.12. Dual bases 18617.13. Jacobian matrix 18717.14. Area of a surface, independence of partition 18817.15. Conformal equivalence 18917.16. Geodesic equation 19017.17. Closed geodesic 191

8 CONTENTS

17.18. Existence of closed geodesic 19217.19. Surfaces of constant curvature 19217.20. Real projective plane 19217.21. Simple loops for surfaces of positive curvature 19317.22. Successive minima 19417.23. Flat surfaces 19517.24. Hyperbolic surfaces 19517.25. Hyperbolic plane 196

Chapter 18. Elements of the topology of surfaces 19918.1. Loops, simply connected spaces 19918.2. Orientation on loops and surfaces 19918.3. Cycles and boundaries 20018.4. First singular homology group 20118.5. Stable norm in 1-dimensional homology 20218.6. The degree of a map 20318.7. Degree of normal map of an embedded surface 20418.8. Euler characteristic of an orientable surface 20418.9. Gauss-Bonnet theorem 20518.10. Change of metric exploiting Gaussian curvature 20518.11. Gauss map 20618.12. An identity 20718.13. Stable systole 20818.14. Free loops, based loops, and fundamental group 20818.15. Fundamental groups of surfaces 209

Chapter 19. Pu’s inequality 21119.1. Hopf fibration h 21119.2. Tangent map 21119.3. Riemannian submersion 21219.4. Riemannian submersions 21219.5. Hamilton quaternions 21319.6. Complex structures on the algebra H 214

Chapter 20. Approach using energy-area identity 21520.1. An integral-geometric identity 21520.2. Two proofs of the Loewner inequality 21620.3. Hopf fibration and the Hamilton quaternions 21820.4. Double fibration of SO(3) and integral geometry on S2 21920.5. Proof of Pu’s inequality 22120.6. A table of optimal systolic ratios of surfaces 221

Chapter 21. A primer on surfaces 223

CONTENTS 9

21.1. Hyperelliptic involution 22321.2. Hyperelliptic surfaces 22521.3. Ovalless surfaces 22521.4. Katok’s entropy inequality 227

Bibliography 231

CHAPTER 1

Analytic geometry

After dealing with classical geometric preliminaries including thetheorema egregium of Gauss, we present new geometric inequalities onRiemann surfaces, as well as their higher dimensional generalisations.

We will first review some familiar objects from classical geometryand try to point out the connection with important themes in modernmathematics.

1.1. Course notes

These notes are based on a course taught at Bar Ilan University; seehttp://u.cs.biu.ac.il/~katzmik/88-526.html (one can also go tohttp://u.cs.biu.ac.il/~katzmik/88-201.html which will redirectto the previous link).

Remark 1.1.1. This year we have an excellent metargel. In pre-vious years, there were various chovrot in hebrew circulating but theycontain serious errors. In some cases there is a direct relationship be-tween the errors contained in the chovrot and mistakes made on theexam. I strongly recommend following these online class notes. Ifthere are any difficulties with the English I would be happy to providea clarification.

The familiar unit circle S1 in the plane, defined to be the locus ofthe equation

x2 + y2 = 1

in the (x, y)-plane. The circle solves the isoperimetic problem in theplane. Namely, consider simple (non-self-intersecting) closed curves ofequal perimeter, for instance a polygon.1

Among all such curves, the circle is the curve that encloses thelargest area. In other words, the circle satisfies the boundary case ofequality in the following inequality, known as the isoperimetric inequal-ity. Let L be the length of the Jordan curve and A the area of the finiteregion bounded by the curve.

1metzula

11

http://u.cs.biu.ac.il/~katzmik/88-526.html

http://u.cs.biu.ac.il/~katzmik/88-201.html

12 1. ANALYTIC GEOMETRY

Theorem 1.1.2 (Isoperimetric inequality). Every Jordan curve inthe plane satisfies the inequality

(L

2π

)2

− A

π≥ 0,

with equality if and only if the curve is a round circle.2

The familiar 2-sphere S2 is a surface that can be defined as thecollection of unit vectors in 3-sphace:

S2 ={(x, y, z) ∈ R3 | x2 + y2 + z2 = 1

}.

Definition 1.1.3. A great circle is the intersection of the spherewith a plane passing through the origin.

The equator is an example of a great circle.

Definition 1.1.4. The great circle distance d(p, q) on S2 is the dis-tance measured along the arcs of great circles connecting the points p, q ∈S2:

d(p, q) = arccos(p · q),where p · q is the usual dot product in Euclidean space.

Namely, the distance between a pair of points p, q ∈ S2 is the lengthof the smaller of the two arcs of the great circle passing through p and q.

In relativity theory, one uses a framework similar to classical differ-ential geometry, with a technical difference having to do with the basicquadratic form being used. Nonetheless, some of the key concepts, suchas geodesic and curvature, are common to both approaches.

In the first approximation, one can think of relativity theory as thestudy of 4-manifolds with a choice of a “light cone”3 at every point.

Einstein gave a strong impetus to the development of differentialgeometry, as a tool in studying relativity. We will systematically useEinstein’s summation convention; see Section 1.3.

2The round circle is the subject of Gromov’s filling area conjecture. The Rie-mannian circle of length 2π is a great circle of the unit sphere, equipped with thegreat-circle distance. The emphasis is on the fact that the distance is measuredalong arcs rather than chords (straight line intervals). For all the apparent simplic-ity of the the Riemannian circle, it turns out that it is the subject of a still-unsolvedconjecture of Gromov’s, namely the filling area conjecture. A surface with a sin-gle boundary circle will be called a filling of that circle. We now consider fillingsof the Riemannian circles such that the ambient distance does not diminish thegreat-circle distance (in particular, filling by the unit disk is not allowed). M. Gro-mov conjectured that Among all fillings of the Riemannian circles by a surface, thehemisphere is the one of least area.

3konus ha’or

1.2. LINEAR ALGEBRA, INDEX NOTATION 13

1.2. Linear algebra, index notation

Linear algebra provides an indispensable foundation for our sub-ject. It is important to develop a facility with Einstein summationconvention (treated in detail in Section 1.3) which will be exploitedthroughout the course.

Let Rn denote the Euclidean n-space. Its vectors will be denoted

v, w ∈ Rn.

Definition 1.2.1. A vector in Rn is a column vector v. To obtaina row vector we take the transpose, vt.

Let B ∈ Mn,n(R) be an n by n matrix. There are two ways ofviewing such a matrix, either as

(1) a linear map (see Remark 1.2.2) or(2) a bilinear form (see Section 1.5).

Developing suitable notation to capture this distinction helps simplifydifferential-geometric formulas down to readable size, and also to mo-tivate the important distinction between a vector and a covector.

Remark 1.2.2 (Matrix as a bilinear form B(v, w)). Consider abilinear form

B(v, w) : Rn × Rn → R, (1.2.1)

sending the pair of vectors (v, w) to the real number vtBw. Here vt isthe row vector given by transpose of v. We write

B = (bij)i=1,...,n; j=1,...,n

so that bij is an entry while (bij), with parentheses, denotes the matrix.

Example 1.2.3. In the 2 by 2 case, we have

v =

(v1

v2

), w =

(w1

w2

), B =

(b11 b12b21 b22

).

Then

Bw =

(b11 b12b21 b22

)(w1

w2

)=

(b11w

1 + b12w2

b21w1 + b22w

2

).

Now the transpose vt = (v1 v2) is a row vector, so

B(v, w) = vtBw

= (v1 v2)

(b11w

1 + b12w2

b21w1 + b22w

2

)

= b11v1w1 + b12v

1w2 + b21v2w1 + b22v

2w2,

and therefore vtBw =∑2

i=1

∑2j=1 bijv

iwj.


We would like to simplify this notation by suppressing the summa-tion symbols “Σ”, as in Section 1.3.

1.3. Einstein summation convention

The following useful notational device was originally introduced byAlbert Einstein.

Definition 1.3.1. The rule is that whenever a product contains asymbol with a lower index and another symbol with the same upperindex, take summation over this repeated index.

Example 1.3.2. Using this notation, the bilinear form (1.2.1) de-fined by the matrix B can be written as follows:

B(v, w) = bijviwj,

with implied summation over both indices.

When we wish to consider a specific term such as ai times vi ratherthan the sum over a dummy index i, we use the underline notation asfollows.

Definition 1.3.3. The expression

aivi

denotes a specific term with index value i (rather than summation overa dummy index i).

1.4. Quadratic forms, polarisation formula

Definition 1.4.1. Let B be a symmetric matrix. The associ-ated quadratic form Q is a quadratic form associated with a bilinearform B(v, w) by the following rule:

Q(v) = B(v, v).

Given v ∈ Rn, let v = viei where {ei}i=1,...,n = {e1, . . . , en} is thestandard basis of Rn.

Lemma 1.4.2. The associated quadratic form satisfies

Q(v) = bijvivj.

Proof. To compute Q(v), we must introduce an extra index j :

Q(v) = B(v, v) = B(viei, vjej) = B(ei, ej)v

ivj = bijvivj, (1.4.1)

proving the lemma. �

1.5. MATRIX AS A LINEAR MAP 15

Theorem 1.4.3. The polarisation formula asserts that

B(v, w) =1

4(Q(v + w)−Q(v − w)).

The polarisation formula allows one to reconstruct the bilinear formfrom the quadratic form, whenever the characteristic of the field is not 2(as is the case for R). For an application, see Section 14.7.

1.5. Matrix as a linear map

Given a real matrix B, consider the associated map

BR : Rn → Rn, v 7→ Bv.

In order to distinguish this case from the case of the bilinear form, wewill raise the first index of the matrix coefficients. Thus, we write B as

B = (bij)i=1,...,n; j=1,...,n

where it is important to s t a g g e r the indices as follows.

Definition 1.5.1. Staggering the indices meaning that we do notplace j under i as in

bij ,

but, rather, leave a blank space (in the place were j used to be), as in

bij .

Consider vectors v = (vj)j=1,...,n and w = (wi)i=1,...,n. Then theequation w = Bv can be written as a system of n scalar equations,

wi = bijvj for i = 1, . . . , n

using the Einstein summation convention (here the repeated index is j).

Definition 1.5.2. The formula for the trace Tr(B) = b11 + b22 +· · ·+ bnn in Einstein notation becomes

Tr(B) = bii

(here the repeated index is i).

Remark 1.5.3. If we wish to specify the i-th diagonal coefficientof our matrix B we use the underline notation

bii

as in Definition 1.3.3.


1.6. Symmetrisation and skew-symmetrisation

For the purposes of dealing with symmetrisation and antisymmetri-sation of a matrix, it is convenient to return to the framework of bilinearforms (i.e., both indices are lower indices).

Definition 1.6.1. The transpose Bt of a matrix B = (bij) is thematrix whose (i, j)-th coefficient is bji.

Geometrically the passage from B to Bt corresponds to reflectionin the main diagonal of the matrix B.

Definition 1.6.2. Let B = (bij). Its symmetric part S is by defi-nition

S =B +Bt

2=(bij + bji

2

)i=1,...,nj=1,...,n

,

while the skew-symmetric part A is

A =B − Bt

2=(bij − bji

2

)i=1,...,nj=1,...,n

.

Theorem 1.6.3. We have B = S + A.

Another useful notation is that of symmetrisation and antisym-metrisation.

Definition 1.6.4. Symmetrisation is defined by setting

b{ij} =1

2(bij + bji) (1.6.1)

and antisymmetrisation by setting

b[ij] =1

2(bij − bji). (1.6.2)

Lemma 1.6.5. A matrix B = (bij) is symmetric if and only if forall indices i and j one has b[ij] = 0.

Proof. We have b[ij] = 12(bij − bji) = 0 since symmetry of B

means bij = bji. �

1.7. Matrix multiplication in index notation

The usual way to define matrix multiplication is as follows. Atriple of matrices A = (aij), B = (bij), and C = (cij) satisfy theproduct relation C = AB if, introducing an additional dummy index k

(cf. formula (1.4.1)), we have the relation cij =∑

k

aikbkj .

1.8. TWO TYPES OF INDICES: SUMMATION INDEX AND FREE INDEX 17

Example 1.7.1 (Skew-symmetrisation of matrix product). By com-mutativity of multiplication of real numbers, we have

aikbkj = bkjaik.

Then the coefficients c[ij] of the skew-symmetrisation of the matrix C =AB satisfy

c[ij] =∑

k

bk[jai]k.

Here by definition

bk[jai]k =1

2(bkjaik − bkiajk).

This notational device will be particularly useful in writing downthe theorema egregium (see Section 10.9). Given below are a few ex-amples of symmetrisation and antisymmetrisation notation:

• See Section 6.3, where we will use formulas of type

gmjΓmik + gmiΓ

mjk = 2gm{jΓ

mi}k;

• See Section 9.6 for Li[jLkℓ];

• See Section 10.8 for Γki[jΓnℓ]m.

The index notation we have described reflects the fact that the nat-ural products of matrices are the ones which correspond to compositionof maps.

Theorem 1.7.2. If A = (aij), B = (bij), and C = (cij) then theproduct relation C = AB corresponding to the composition of linearmaps CR = AR ◦BR simplifies to the relation

cij = aikbkj for all i, j. (1.7.1)

The proof is immediate from the definition of matrix multiplication.

1.8. Two types of indices: summation index and free index

In expressions of type (1.7.1) it is important to distinguish betweentwo types of indices: a free index or an summation index.

Definition 1.8.1. An index appearing both as a subscript and asuperscript is called an summation index. The remaining indices arecalled free.

Example 1.8.2. In formula (1.7.1) the index k is a summationindex whereas indices i and j are free indices.


1.9. The inverse matrix, Kronecker delta

The Kronecker delta function δ defined as follows.

Definition 1.9.1. The expression

δij =

{1 if i = j0 if i 6= j

is called the Kronecker delta function.

Consider a matrix B = (bij).

Definition 1.9.2. The inverse matrix B−1 is the matrix

B−1 = (bij)i=1,...,n; j=1,...,n

where both indices have been raised.

Then the equation B−1B = I becomes

bikbkj = δij (1.9.1)

in Einstein notation with repeated index k.

Remark 1.9.3. In (1.9.1) the index k is a summation index whereas iand j are free indices.

Example 1.9.4. The identity endomorphism I = (δij) by definition

satisfies AI = A = IA for all endomorphisms A = (aij), or equivalently

aijδjk = aik = δija

jk, (1.9.2)

using the Einstein summation convention.

Remark 1.9.5. In expression (1.9.2) the index j is a summationindex whereas indices i and k are free indices.

Example 1.9.6. Let δij be the Kronecker delta function on Rn,where i, j = 1, . . . , n, viewed as a linear transformation Rn → Rn.

(1) Evaluate the expression δijδjk paying attention to which the

summation indices are;(2) Evaluate the expression δijδ

ji paying attention to which the

summation indices are.

Definition 1.9.7. Given a pair of vectors v = viei and w = wjejin R3, their vector product is a vector v ×w ∈ R3 satisfying one of thefollowing two equivalent conditions:

1.10. EIGENVALUES, SYMMETRY 19

(1) we have v × w = det

e1 e2 e3v1 v2 v3

w1 w2 w3

, in other words,

v × w = (v2w3 − v3w2)e1 − (v1w3 − v3w1)e2 + (v1w2 − v2w1)e3

= 2(v[2w3]e1 − v[1w3]e2 + v[1w2]e3

).

(2) the vector v × w is perpendicular to both v and w, of lengthequal to the area of the parallelogram spanned by the two vec-tors, and furthermore satisfying the right hand rule,4 mean-ing that the 3 by 3 matrix formed by the three vectors v, w,and v × w has positive determinant.

Remark 1.9.8. We have an identity

a× (b× c) = (a · c)b− (a · b)c (1.9.3)

for every triple of vectors a, b, c in R3. Note that the expression (1.9.3)vanishes if a is perpendicular to both b and c.

1.10. Eigenvalues, symmetry

Properly understanding surface theory and related key conceptssuch as the Weingarten map (see Section 8.10) depends on linear-algebraic background related to diagonalisation of symmetric matricesor, more generally, selfadjoint endomorphisms.

In general, a real matrix may have no real eigenvector or eigenvalue.

Example 1.10.1. The matrix of a 90 degree rotation in the plane,(0 −11 0

)

does not have a real eigenvector for obvious geometric reasons (nodirection is preserved but rather rotated).

In this section we prove the existence of a real eigenvector (andhence, a real eigenvalue) for a real symmetric matrix.

This fact has important ramifications in surface theory, since thevarious notions of curvature of a surface are defined in terms of theeigenvalues of a certain selfadjoint operator (see sections 8.6 and 10.1)which is an invariant formulation of the notion of a symmetric matrix.Let

I =

1 0

. . .0 1

4Klal yad yamin


be the (n, n) identity matrix. Thus

I = (δij) i=1,...,nj=1,...,n

where δij is the Kronecker delta. Let B be an (n, n)-matrix.

Definition 1.10.2. A real number λ is called an eigenvalue of Bif

det(B − λI) = 0.

Theorem 1.10.3. If λ ∈ R is an eigenvalue of B, then there is avector v ∈ Rn, v 6= 0, such that

Bv = λv. (1.10.1)

The proof is one of the first results in linear algebra and is notreproduced here.

Definition 1.10.4. A nonzero vector satisfying (1.10.1) is calledan eigenvector belonging to λ.

1.11. Euclidean inner product

Definition 1.11.1. The Euclidean inner product of vectors v, w isdefined by

〈v, w〉 = v1w1 + · · ·+ vnwn =n∑

i=1

viwi.

Recall that all of our vectors are column vectors.

Lemma 1.11.2. The inner product can be expressed in terms ofmatrix multiplication in the following fashion: 〈v, w〉 = vt w.

Recall the following theorem from basic linear algebra.

Theorem 1.11.3. The transpose has the following property:

(AB)t = BtAt.

Definition 1.11.4. A square matrix A is called symmetric of At =A.

Lemma 1.11.5. A real matrix B is symmetric if and only if forall v, w ∈ Rn, one has

〈Bv,w〉 = 〈v, Bw〉.Proof. We have

〈Bv,w〉 = (Bv)tw = vtBtw = 〈v, Btw〉 = 〈v, Bw〉,proving the =⇒ direction. Conversely, if v = ri and w = ejthen 〈Bv,w〉 = bij while 〈v, Bw〉 = bji, proving the ⇐= direction. �

CHAPTER 2

Eigenvalues of symmetric matrices, conic sections

2.1. Finding an eigenvector of a symmetric matrix

In this section we continue with linear-algebraic preliminaries forthe theory of surfaces in Euclidean space.

Theorem 2.1.1. Every real symmetric matrix possesses a real eigen-vector.

We will give two proofs of this important theorem. The first proofis simpler, more algebraic, and passes via complexification. The secondproof is more geometric.

First proof. Let n ≥ 1, and let B ∈ Mn,n(R) be an n × n realsymmetric matrix. As such, it defines a linear map

BR : Rn → Rn, v 7→ Bv

sending a vector v ∈ Rn to the vector Bv. We now use the fieldextension R → C and view B as a complex matrix B ∈ Mn,n(C).Then the matrix B defines a complex linear map (endomorphism)

BC : Cn → Cn

sending v ∈ Cn to Bv.Let 〈 , 〉 be the standard Hermitian inner product in Cn. Recall

that the Hermitian inner product is linear in one variable and skew-linear in the other. Thus, we have1

〈z, w〉 =n∑

i

ziwi.

Then 〈Bv, v〉 = 〈v, Bv〉 since the matrix B = Bt is real symmetric.The characteristic polynomial pB(λ) of the endomorphism BC is

a polynomial of positive degree n > 0. Therefore pB(λ) possesses aroot λ ∈ C by the fundamental theorem of algebra. Let v ∈ Cn be

1Alternatively, some texts adopt the convention 〈z, w〉 =∑ni=1 ziwi.

21

22 2. EIGENVALUES OF SYMMETRIC MATRICES, CONIC SECTIONS

an eigenvector belonging to λ, so that Bv = λv. Since the Hermitianinner product is skew-symmetric in the second variable, we obtain

〈λv, v〉 = 〈v, λv〉 = λ〈v, v〉.Therefore λ〈v, v〉 = λ〈v, v〉 so that λ = λ, i.e., λ is a real eigenvalue, asrequired. �

The second proof is somewhat longer but has the advantage of beingmore geometric, as well as more concrete in the choice of the vector inquestion.

Second Proof. Let S ⊆ Rn be the unit sphere S = {v ∈ Rn : ‖v‖ =1}. Given a symmetric matrix B, define a function f : S → R as the re-striction to S of the function, also denoted f , given by f(v) = 〈v,Bv〉.Let v0 be a maximum of f restricted to S. Let V ⊥

0 ⊆ Rn be the orthog-onal complement of the line spanned by v0. Let w ∈ V ⊥

0 . Consider thecurve v0 + tw, t ≥ 0 (see also a different choice of curve in Remark 2.1.2below). Then d

dt

∣∣t=0

f(v0 + tw) = 0 since v0 is a maximum and w is tangentto the sphere. Now

d

dt

∣∣∣0f(v0 + tw) =

d

dt

∣∣0〈v0 + tw,B(v0 + tw)〉

=d

dt

∣∣0(〈v0, Bv〉+ t〈v0, Bw〉+ t〈w,Bv0〉+ t2〈w,Bv〉)

= 〈v0, Bw〉+ 〈w,Bv0〉= 〈Bw, v0〉+ 〈Bv0, w〉= 〈Btv0, w〉+ 〈Bv0, w〉= 〈(Bt +B)v0, w〉= 2〈Bv0, w〉 by symmetry of B.

Thus 〈Bv0, w〉 = 0 for all w ∈ V ⊥0 . Hence Bv0 is proportional to v0 and

so v0 is an eigenvector of B. �

Remark 2.1.2. Our calculation used the curve v0 + tw which, whiletangent to S at v0 (see section 6.9), does not lie on S. If one prefers, onecan use instead the curve (cos t)v0 + (sin t)w lying on S. Then

d

dt

∣∣∣t=0

〈(cos t)v0 + (sin t)w,B((cos t)v0 + (sin t)w)〉 = · · · = 〈(Bt +B)v0, w〉and one argues by symmetry as before.

2.2. Trace of product in index notation

First we reinforce the material on index notation and Einstein sum-mation convention; see Section 1.3. The following result is importantin its own right. We reproduce it here because its proof is a goodillustration of the uses of the Einstein index notation.

2.4. ORTHOGONAL DIAGONALISATION OF SYMMETRIC MATRICES 23

Theorem 2.2.1. Let A and B be square n × n matrices. Thentr(AB) = tr(BA).

Proof. Let A = (aij) and B = (bij). Then

tr(AB) = tr(aikbkj) = aikb

ki

by definition of trace (see Definition 1.5.2). Meanwhile,

tr(BA) = tr(bkiaij) = bkia

ik = aikb

ki = tr(AB),

proving the theorem. �

Exercise 2.2.2. A matrix A is called idempotent if A2 = A. Writethe idempotency condition in indices with Einstein summation conven-tion (without Σ’s), keeping track of free indices and internal summationindices.

Exercise 2.2.3. Matrices A and B are similar if there exists aninvertible matrix P such that AP − PB = 0. Write the similaritycondition in indices, as the vanishing of each (i, j)th coefficient of thedifference AP − PB.

2.3. Inner product spaces and self-adjoint operators

In Section 2.1 we worked with real matrices and showed that thesymmetry of a matrix guarantees the existence of a real eigenvector.In a more general situation where a natural basis is not available, asimilar statement holds for a special type of endomorphism of a realvector space.

Definition 2.3.1. Let (V, 〈 , 〉) be a real inner product space. Anendomorphism B : V → V is called selfadjoint if one has

〈Bv,w〉 = 〈v, Bw〉 ∀v, w ∈ V. (2.3.1)

Corollary 2.3.2. Every selfadjoint endomorphism of a real innerproduct space admits a real eigenvector.

Proof. The selfadjointness was the relevant property in the proofof Theorem 2.1.1. �

2.4. Orthogonal diagonalisation of symmetric matrices

Consider an endomorphism E : V → V .

Definition 2.4.1. A subspace U ⊆ V is invariant under E ifE(U) ⊆ U .


Definition 2.4.2. The orthogonal complement O ⊆ V of a sub-space U ⊆ V is defined by O = {x ∈ V : 〈x, u〉 = 0 ∀u ∈ U}, writtenas V = U +O.

Theorem 2.4.3. If U and O are orthogonal complements in Vthen dimV = dimU + dimO.

This is known from linear algebra.

Example 2.4.4. The orthogonal complement of the plane

{(x, y, z) : ax+ by + cz = 0}in R3 is the line spanned by the vector (a, b, c)t.

Lemma 2.4.5. Let E : V → V be a selfadjoint endomorphism ofan inner product space V and let U ⊆ V be an E-invariant subspace.Then the orthogonal complement of U in V is also E-invariant.

Proof. Let w be orthogonal to U ⊆ V . Let u ∈ U be an arbitraryvector. Then

〈E(w), u〉 = 〈w,Eu〉 = 0

since E(u) ∈ U by E-invariance. Therefore E(w) is also orthogonal tothe vector u. Since this is valid for each u ∈ U , the vector w belongsto the orthogonal complement of U , as required. �

Theorem 2.4.6. Every real symmetric matrix can be orthogonallydiagonalized.

Proof. A symmetric n × n matrix S ∈ Mn,n(R) defines a selfad-joint endomorphism SR of the real inner product space V = Rn givenby

SR : V → V, v 7→ Sv.

By Corollary 2.3.2, every selfadjoint endomorphism has a real eigen-vector v1 ∈ V , which we can assume to be a unit vector:

|v1| = 1.

Let λ1 ∈ R be its eigenvalue. Now we let V1 = V and set

V2 ⊆ V1

be the orthogonal complement of the line Rv1 ⊆ V1. Thus we have anorthogonal decomposition

V1 = Rv1 + V2.

by Lemma 2.4.5, the subspace V2 is invariant under SR. The restriction

SR⇂V2

2.5. CLASSIFICATION OF CONIC SECTIONS: DIAGONALISATION 25

of SR to V2 is still selfadjoint by inheriting the property (2.3.1). Namely,since property (2.3.1) holds for all vectors v, w ∈ V , it still holds if thesevectors are restricted to vary in a subspace V2 ⊆ V , i.e.,

〈SRv, w〉 = 〈v, SRw〉 ∀v, w ∈ V2. (2.4.1)

Therefore we can apply Corollary 2.3.2 to the selfadjoint endomor-phism SR⇂V2 . Thus we can find an eigenvector v2 ∈ V2 of SR⇂V2 witheigenvalue λ2 ∈ R. Let V3 ⊆ V2 be the orthogonal complement of theline Rv2 spanned by v2. Choose an eigenvector v3 ∈ V3, etc. Arguinginductively, we obtain a strictly decreasing sequence of spaces

V1 ⊃ V2 ⊃ V3 ⊃ . . . ⊃ Vi ⊃ . . .

which must end with a point since V1 is finite-dimensional. We thusobtain an orthonormal basis consisting of eigenvectors v1, . . . , vn ∈ Vwith eigenvalues λ1, . . . , λn ∈ R. Let

P = [v1 . . . vn]

be the orthogonal n × n matrix whose columns are the vectors vi, sothat we have

P−1 = P t. (2.4.2)

Consider the diagonal matrix Λ = diag(λ1, . . . , λn). By construction,we have

S = PΛP t

from (2.4.2), or equivalently,

SP = PΛ.

Indeed, to verify the relation SP = PΛ, note that both sides are equalto the square matrix [λ1v1 λ2v2 . . . λnvn]. �

2.5. Classification of conic sections: diagonalisation

Definition 2.5.1. A conic section2 (or conic for short) in the planeis by definition a curve defined by the following master equation inthe (x, y)-plane:

ax2 + 2bxy + cy2 + dx+ ey + f = 0, a, b, c, d, e, f ∈ R. (2.5.1)

Here we chose the coefficient of the xy term to be 2b rather than bso as to simplify formulas like (2.5.2) below.

Example 2.5.2. We have the following examples:

(1) a circle x2 + y2 − 1 = 0,(2) an ellipse x2 + 3y2 − 1 = 0,

2chatach charut


(3) a parabola x2 + y = 0,(4) a hyperbola 2xy + 1 = 0,(5) a hyperbola x2 − 5y2 − 1 = 0.

With reference to the master equation (2.5.1) we now let X =

(xy

)

and let

S =

(a bb c

). (2.5.2)

Then X tSX = ax2 + 2bxy + cy2.

Theorem 2.5.3. Up to an orthogonal transformation resulting innew coordinates (x′, y′), every conic section as in (2.5.1) can be writtenin a “diagonal” form

λ1x′2 + λ2y

′2 + d′x′ + e′y′ + f = 0, (2.5.3)

where the coefficients λ1 and λ2 are the eigenvalues of the matrix Sof (2.5.2).

Proof. Consider the row vector

T =(d e

).

Then TX = dx+ ey. Thus equation (2.5.1) becomes

X tSX + TX + f = 0. (2.5.4)

We now apply Theorem 2.4.6 to orthogonally diagonalize S to ob-tain S = PΛP t with

Λ =

(λ1 00 λ2

). (2.5.5)

Substituting this expression for S into (2.5.4) yields

X tPΛP tX + TX + f = 0.

We setX ′ = P tX . Then X = PX ′ since P is orthogonal. Furthermore,we have

(X ′)t = (P tX)t = (X t)(P t)t = (X t)P.

Hence we obtain(X ′)tΛX ′ + TPX ′ + f = 0.

Letting T ′ = TP , we obtain

(X ′)tΛX ′ + T ′X ′ + f = 0,

where Λ is the diagonal matrix of (2.5.5). Letting x′ and y′ be the

components of X ′, i.e. X ′ =

(x′

y′

), we obtain formula (2.5.3), as re-

quired. �

2.6. CLASSIFICATION OF CONICS: TRICHOTOMY, NONDEGENERACY 27

Example 2.5.4. The hyperbola 2xy − 1 = 0 is not in “diagonal”

form. The corresponding matrix S is S =

(0 11 0

). Here the eigenval-

ues are λ1 = 1 and λ2 = −1. By Theorem 2.5.3 we obtain the “diag-onal” equation x′2 − y′2 − 1 = 0 in the new coordinates (x′, y′). The

result could be obtained directly by using the substitution x = x′+y′√2

and y = x′−y′√2.

To obtain more precise information about the conic, we need tospecify certain nondegeneracy conditions, as discussed in Section 2.6.

2.6. Classification of conics: trichotomy, nondegeneracy

We apply the diagonalisation result of Section 2.5 to classify conicsections into three types (under suitable nondegeneracy conditions):ellipse, parabola, hyperbola. Let S be the matrix (2.5.2).

Theorem 2.6.1. Suppose det(S) 6= 0, i.e., S is invertible. Then,up to an orthogonal transformation and a translation, the conic sectioncan be written in the form

λ1(x′′)2 + λ2(y

′′)2 + f ′′ = 0 (2.6.1)

(note that the constant term is changed).

Proof. If the determinant is nonzero then both eigenvalues λi arenonzero. The term d′x′ in (2.5.3) can be absorbed into the quadraticterm λ1x

′2 by completing the square as follows:

λ1x′2 + d′x′ = λ1

(x′2 + 2

d′

2λ1x′)

= λ1

(x′2 + 2

d′

2λ1x′ +

(d′

2λ1

)2)

− λ1

(d′

2λ1

)2

= λ1

(x′ +

d′

2λ1

)2

− d′2

4λ1,

and we set

x′′ = x′ +d′

2λ1.

Similarly e′y′ can be absorbed into λ2y′2. Geometrically this corre-

sponds to a translation along the axes x′ and y′, proving the theo-rem. �

Definition 2.6.2. A conic section of type (2.6.1) is called a hy-perbola if λ1λ2 < 0, provided the following nondegeneracy condition issatisfied: the constant f ′′ in equation (2.6.1) is nonzero.


Corollary 2.6.3 (Degenerate case). Assume det(S) < 0. If theconstant f ′′ is zero, then instead of a hyperbola, the solution set is adegenerate conic given by a pair of transverse lines.

Remark 2.6.4. The transverse lines as in Corollary 2.6.3 are or-thogonal if and only if λ1 = −λ2.

Definition 2.6.5. A conic section is called an ellipse if λ1λ2 > 0,provided the following nondegeneracy condition is satisfied: the con-stant f ′′ in equation (2.6.1) is nonzero and has the opposite sign ascompared to the sign of λ1, i.e., f

′′λ1 < 0.

See Example 11.6.6 for an application.

Corollary 2.6.6 (Degenerate case). Assume det(S) > 0. If theconstant f ′′ is zero, then instead of an ellipse one obtains a singlepoint x′′ = y′′ = 0. If the constant f ′′ 6= 0 has the same sign as λ1 thenthe solution set is empty.

Recall that the determinant of the matrix S is the product of itseigenvalues: det(S) = ac − b2 = λ1λ2. We summarize our conclusionsfor the ellipse and the hyperbola in the following corollary.

Corollary 2.6.7. We have the following relations between the al-gebraic equation and the nature of the corresponding conic:

(1) If the conic ax2 + 2bxy + cy2 + dx + ey + f = 0 is an ellipsethen ac− b2 > 0.

(2) If ac − b2 > 0 and the solution locus is neither empty nor asingle point, then it is an ellipse.

Corollary 2.6.8. We have the following relations between the al-gebraic equation and the nature of the corresponding conic:

(1) If the conic ax2 + 2bxy + cy2 + dx+ ey + f = 0 is a hyperbolathen ac− b2 < 0.

(2) If ac−b2 < 0 then the solution locus is not a pair of transverselines, then the conic is a hyperbola.

2.7. Characterisation of parabolas

Suppose det(S) = 0. In this case, one cannot necessarily eliminatethe linear terms by completing the square as in the cases of ellipseand hyperbola in Section 2.6. Therefore we continue working with theequation

λ1x′2 + λ2y

′2 + d′x′ + e′y′ + f = 0 (2.7.1)

from (2.5.3) where now one of the λi is zero. This is legitimate becausediagonalizing S does not depend on S being invertible.

2.8. SUMMARY: CLASSIFICATION OF CONICS 29

Definition 2.7.1. The conic is a parabola if the following two con-ditions are satisfied by the coefficients in (2.7.1):

(1) the matrix S is of rank 1 (equivalently, λ1λ2 = 0 and λ1 6= λ2);(2) if λ1 = 0 then d′ 6= 0; if λ2 = 0 then e′ 6= 0.

Corollary 2.7.2 (Degenerate case). Suppose S is of rank 1 andafter absorbing the linear term in (2.7.1) the equation becomes λ1x

′′2+f ′′ = 0 or λ2y

′′2+f ′′ = 0. Then the solution set is either empty, a line,or a pair of parallel lines.

Example 2.7.3. The equation x2 + 1 = 0 has an empty solutionset in the (x, y)-plane. The equation x2 = 0 has as solution set a singleline, namely the y-axis. The equation x2 − 1 = 0 has as solution set apair of parallel lines.

2.8. Summary: classification of conics

The analysis of the cases presented in Sections 2.6 and 2.7 resultsin the following classification.

Theorem 2.8.1. Assume not all of a, b, c are 0. A real conic ax2+2bxy + cz2 + dx + ey + f = 0 is represented, up to congruence andtranslation, by one of the following possible sets:

(1) the empty set ∅;(2) a single point;(3) union of a pair of transverse lines;(4) a single line or a pair of parallel lines;(5) ellipse (ac− b2 > 0);(6) parabola (ac− b2 = 0);(7) hyperbola (ac− b2 < 0).

The first four cases are known as degenerate cases. A parabola canoccur only if ac − b2 = 0. An ellipse can occur only if ac − b2 > 0.A hyperbola can occur only if ac− b2 < 0.

CHAPTER 3

Quadric surfaces, Hessian, representation of curves

3.1. Quadric surfaces

Quadric surfaces are a rich source of examples that will help usillustrate basic notions of differential geometry such as Gaussian cur-vature.

Definition 3.1.1. A quadric surface in R3 is the locus of points (x, y, z)satisfying the master equation

ax2+2bxy+ cy2+2dxz+fz2+2gyz+hx+ iy+ jz+k = 0, (3.1.1)

where a, b, c, d, f, g, h, i, j, k ∈ R.

To bring equation (3.1.1) to standard form, we apply an orthogo-nal diagonalisation procedure similar to that employed in Section 2.6.Thus, we define matrices S, X , and D by setting

S =

a b db c gd g f

, X =

xyz

, T =

(h i j

),

so that the quadratic part of (3.1.1) becomes X tSX , and the linearpart becomes DX . Then equation (3.1.1) takes the form

X tSX +DX + k = 0. (3.1.2)

Theorem 3.1.2. By means of an orthogonal transformation andtranslation, the general equation of a quadric surface can be simplifiedto

λ1x2 + λ2y

2 + λ3z2 + dx+ fy + gz + h = 0, (3.1.3)

with new variables x, y, z and new coefficients λ1, λ2, λ3, d, f, g, h ∈ R.

Proof. We orthogonally diagonalize S as in Section 2.5. �

3.2. Case of eigenvalues (+ + +) or (−−−), ellipsoid

Definition 3.2.1. A quadric surface is an ellipsoid if the coeffi-cients λi, i = 1, 2, 3 in (3.1.3) are nonzero and have the same sign, andmoreover the solution locus is neither a single point nor the empty set.

31

32 3. QUADRIC SURFACES, HESSIAN, REPRESENTATION OF CURVES

Theorem 3.2.2. Suppose the eigenvalues of S in the master equa-tion (3.1.1) of a quadric surface are nonzero and have the same sign.Then an orthogonal transformation and translation of the coordinatesreduce the equation to the form

λ1x2 + λ2y

2 + λ3z2 + ℓ = 0,

resulting in an ellipsoid if λ1ℓ < 0, a degenerate surface given by asingle point when ℓ = 0, and the empty set when λ1ℓ > 0.

Proof. By Theorem 3.1.2 there exists an orthogonal transforma-tion diagonalizing S. Next, we use the nonvanishing of the eigenvaluesto complete the square so as to eliminate the first-order term D inequation (3.1.2), as in Section 2.6. �

Example 3.2.3. We have the following examples of ellipsoids:

(1) the equation 3x2 + 5y2 + 7z2 − 1 = 0 defines an ellipsoid,(2) equivalently −3x2−5y2−7z2+1 = 0 gives the same ellipsoid,(3) the equation 3x2+5y2+7z2 = 0 degenerates to a single point

(the origin),(4) equation 3x2+5y2+7z2+1 = 0 is the empty degenerate conic.

3.3. Determination of type of surface

Consider the surface defined by the equation

3x2 + y2 − 2xz + 3z2 − 5 = 0.

The corresponding symmetric matrix S ∈M3,3(R) is

S =

3 0 −10 1 0−1 0 3

.

We notice that the equation is partly diagonalized already becausethere are no xy or yz terms. Thus the y-axis is an invariant subspaceof S. Similarly, the (x, z)-plane is invariant under S.

Restricting to the (x, z) plane, we obtain the equation

3x2 − 2xz + 3z2 − 1 = 0. (3.3.1)

We therefore focus on diagonalizing (3.3.1). The corresponding sym-metric matrix is

C =

(3 −1−1 3

)

with characteristic polynomial pC(λ) = λ2 − 6λ + 8. Its roots are 3 ±√9− 8 = 3 ± 1. Both roots 2 and 4 are positive. With respect to

3.4. CASE OF EIGENVALUES (+ +−) OR (+−−), HYPERBOLOID 33

the new coordinates x′, z′ we therefore obtain the quadratic expres-sion 2x′2 + 4z′2. Here there is no need to calculate x′, z′ explicitly.

In the new coordinates, the equation of the quadric surface takesthe form 2x′+y2+4z′2−5 = 0. Notice that all three eigenvalues 1, 2, 4of the original matrix S are positive, and the solution set is neither apoint nor the empty set. By Theorem 3.2.2, the surface is an ellipsoid.

3.4. Case of eigenvalues (+ +−) or (+−−), hyperboloid

Assume λ1λ2λ3 6= 0.

Definition 3.4.1. The equation

λ1x2 + λ2y

2 + λ3z2 = 0 (3.4.1)

represents a degenerate case called a cone when λ1, λ2, λ3 include bothpositive and negative values. A surface defined by an equation reducibleto (3.4.1) by an orthogonal transformation is similarly called a cone.

Example 3.4.2. The equation x2 + y2− z2 = 0 defines a cone, andsimilarly the equation −x2 − y2 + z2 = 0 defines (the same) cone.

Definition 3.4.3. A quadric surface is a hyperboloid if the coef-ficients λ1, λ2, λ3 in (3.1.3) include both positive and negative values,and the surface is not a cone.

Theorem 3.4.4. Suppose det(S) 6= 0 in the master equation of aquadric surface.

(1) If all eigenvalues have the same sign and the surface is neitherthe empty set nor single point, then the surface is an ellipsoid;

(2) if the eigenvalues include both positive and negative values andthe surface is not a cone, then the surface is a hyperboloid.

Proof. As before. �

Definition 3.4.5. The hyperboloid of one sheet1 is the locus of theequation az2 = bx2 + cy2 − d where a > 0, b > 0, c > 0, d > 0. TheGaussian curvature is negative at each point.

Definition 3.4.6. The hyperboloid of two sheets2 is the locus ofthe equation az2 = bx2 + cy2 + d where a > 0, b > 0, c > 0, d > 0. TheGaussian curvature is positive at each point.

1chad-yeriati2du-yeriati


3.5. Rank 2, hyperboloid, cylinder

We have rank(S) = 2 if exactly two of the three eigenvalues arenonzero. We study quadric surfaces X tSX +DX + k = 0 in the casewhen S has rank 2.

Theorem 3.5.1. Suppose rank(S) = 2. Up to orthogonal transfor-mation and translation, the quadric surface X tSX+DX+k = 0 takesthe form

λ1x2 + λ2y

2 + cz + d = 0, (3.5.1)

with new variables x, y, z and new coefficients, where λ1 6= 0 and λ2 6=0.

Proof. As a first step, we orthogonally diagonalize the matrixas before, and relabel the new variables x, y, z in such a way that theeigendirections for the nonzero eigenvalues correspond to the x-axis andthe y-axis. Next, we eliminate the linear term in x and y by absorbing itinto the respective quadratic term by completing the square as before.Note that the linear term in z cannot be eliminated because there isno quadratic term since the third eigenvalue vanishes. �

Definition 3.5.2. If c 6= 0 in equation (3.5.1), the correspondingnondegenerate quadric surface is called a paraboloid.

Definition 3.5.3. Suppose c = 0 in equation (3.5.1), resulting inequation λ1x

2 + λ2y2 + d = 0. All such surfaces are called degenerate.

Example 3.5.4. The cylinder x2 + y2 − 1 = 0 is an example of adegenerate quadric surface.

Additional special cases of quadric surfaces are the following. It isimportant to think through each of these examples as they will provideimportant illustrations of the behavior of the Gaussian curvature (tobe introduced in Section 9.6) of surfaces.

Example 3.5.5. The (elliptic) paraboloid z = ax2 + by2, ab > 0.The Gaussian curvature is positive at each point.

Example 3.5.6. The hyperbolic paraboloid z = ax2 − by2, ab > 0.The Gaussian curvature is negative at each point.

Theorem 3.5.7. Consider the surface

M = {(x, y, z) ∈ R3 : z = ax2 + by2} where ab 6= 0.

If a and b have the same sign then the surface is a paraboloid. If aand b have opposite sign then the surface is a hyperbolic paraboloid.

This is immediate from the discussion above.

3.5. RANK 2, HYPERBOLOID, CYLINDER 35

Remark 3.5.8. In Section 2.8 we gave a (more-or-less) completeclassification of conic sections, including the degenerate cases. Forquadric surfaces, a complete classification in the case det(S) = 0 is toodetailed to be treated here.

3.5.1. Jacobi’s criterion. This section is optional. The type of quadricsurface one obtains depends critically on the signs of the eigenvalues of thematrix S. The signs of the eigenvalues can be determined without diago-nalisation by means of Jacobi’s criterion. Given a matrix A over a field F ,let ∆k denote the k × k upper-left block, called a principal minor. Ma-trices A and B are equivalent if they are congruent (rather than similar),meaning that B = P tAP (rather than by conjugation P−1AP ). Equivalentmatrices don’t have the same eigenvalues (unlike similar matrices). Here Bis of course not assumed to be orthogonal.

Theorem 3.5.9 (Jacobi). Let A ∈ Mn(F ) be a symmetric matrix, andassume det(∆k) 6= 0 for k = 1, . . . , n. Then A is equivalent to the matrix

diag(

1∆1, ∆1

det∆2, . . . , det∆n−1

det∆n

).

For example, for a 2 × 2 symmetric matrix

(a bb d

)Jacobi’s criterion

affirms the equivalence to

(1a 00 a

ad−b2

).

Proof. Take the vector bk = ∆−1k ek ∈ F k of length k, and pad it with

zeros up to length n. Consider the matrix B = (bij) whose column vectorsare b1, . . . , bn. By Cramer’s formula, the diagonal coefficients of B sat-

isfy bkk = det

(∆k−1 00 1/det∆k

)= det∆k−1/∆k, so det(B) =

∏nk=1 bkk =

1/det(A) 6= 0. Compute that BtAB is lower triangular with diagonalb11, . . . , bkk. Being symmetric, it is diagonal. �

If some minor ∆k is not invertible, then A cannot be definite. Applyingthis result in the case of a real symmetric matrix, we obtain the followingcorollary. Let A be symmetric. Then A is positive definite if and only if alldet(∆k) > 0.

Define minors in general (choose rows and columns i1, . . . , it). Permutingrows and columns, we obtain the following corollary.

Corollary 3.5.10. Let A be symmetric positive definite matrix. Thenall “diagonal” minors are positive definite (and in particular have positivedeterminants).

An application of this is determining whether or not a quadric surfaceis an ellipsoid, without having to orthogonally diagonalize the matrix ofcoefficients, as we will see below (care has to be taken to show that the


quadric surface is nondegenerate). For example, let us determine whetheror not the quadric surface

x2 + xy + y2 + xz + z2 + yz + x+ y + z − 2 = 0 (3.5.2)

is an ellipsoid. To solve the problem, we first construct the corresponding

matrix S =

1 1/2 1/21/2 1 1/21/2 1/2 1

and then calculate the principal minors ∆1 =

1, ∆2 = 1 · 1− 12 · 1

2 = 3/4, and

∆3 = 1 +1

8+

1

8− 1

4− 1

4− 1

4=

1

2.

Thus all principal minors are positive and therefore the surface is an ellip-

soid, provided we can show it is nondegenerate. To check nondegeneracy,

notice that (3.5.2) has at least two distinct solutions: (x, y, z) = (1, 0, 0)

and (x, y, z) = (0, 1, 0). Therefore it is a nondegenerate ellipsoid.

3.6. Exercise on index notation

Theorem 3.6.1. Every 2×2 matrix A = (aij) satisfies the identity

aikakj + det(A) δij = akka

ij ∀i, j (3.6.1)

Note that indices i and j and free indices, whereas k is a summationindex (see Section 1.8).

Proof. We will use the Cayley-Hamilton theorem which assertsthat pA(A) = 0 where pA(λ) is the characteristic polynomial of A. Inthe rank 2 case, we have pA(λ) = λ2 − (trA)λ + det(A), and thereforewe obtain

A2 − (trA)A+ det(A)I = 0,

which in index notation gives aikakj − akka

ij + detA δij = 0. This is

equivalent to (3.6.1). �

Example 3.6.2 (Exercise on index notation). For a matrix A ofsize 3 × 3, the characteristic polynomial pA(λ) has the form pA(λ) =λ3 − Tr(A)λ2 + s(A)λ − det(A)λ0. Here s(A) = λ1λ2 + λ1λ3 + λ2λ3,where λi are the eigenvalues of A. By Cayley-Hamilton theorem, wehave

pA(A) = 0. (3.6.2)

Express the equation (3.6.2) in index notation.

3.7. HESSIAN, MINIMA, MAXIMA, SADDLE POINTS 37

3.7. Hessian, minima, maxima, saddle points

Let (u1, . . . , un) be coordinates in Rn. Consider a C2-smooth func-tion f(u1, . . . , un) of n variables.

Definition 3.7.1. The gradient of f at a point p = (u1, . . . , un) isthe vector

∇f(p) =

∂f∂u1∂f∂u2...∂f∂un

Definition 3.7.2. A critical point p ∈ Rn of f is a point satisfying

∇f(p) = 0,

i.e. ∂f∂ui

(p) = 0 for all i = 1, . . . , n.

We now consider the second partial derivatives of f , denoted fij =∂2f

∂ui∂uj.

Definition 3.7.3. the Hessian matrix of f

Hf = (fij)i=1,...,n; j=1,...,n

is the n× n matrix of second partials.

The antisymmetrisation notation was defined in formula (1.6.2).We can then formulate what is known as Schwarz’s theorem or Clairaut’stheorem as follows.

Theorem 3.7.4 (Equality of mixed partials). Let f ∈ C2. Thefollowing three statements are equivalent and true:

(1) the Hessian Hf is a symmetric matrix;(2) in terms of the antisymmetrisation notation, f[ij] = 0;(3) we have fij = fji for all i, j.

Example 3.7.5 (maxima, minima, saddle points). Let n = 2. Thenthe sign of the determinant

det(Hf) =∂2f

∂u1∂u1∂2f

∂u2∂u2−(

∂2f

∂u1∂u2

)2

at a critical point has geometric significance. Namely, if

det(Hf(p)) > 0,

then p is a local maximum or a local minimum. If det(Hf(p)) < 0,then p is a saddle point.3

3Nekudat ukaf


Example 3.7.6 (Quadric surfaces). Quadric surfaces are a richsource of examples.

(1) The origin is a critical point for the function whose graph isthe paraboloid z = x2 + y2. In the case of the paraboloid thecritical point is a minimum.

(2) Similar remarks apply to the top sheet of the hyperboloid of

two sheets, namely z =√x2 + y2 + 1, where we also get a

minimum.(3) The origin is a critical point for the function whose graph is

the hyperbolic paraboloid z = x2 − y2. In the case of thehyperbolic paraboloid the critical point is a saddle point.

In addition to the sign, the value of Hf(p) also has geometric signif-icance in terms of an invariant we will define later called the Gaussiancurvature, expressed by the following theorem.

Theorem 3.7.7. Let f be a function of two variables. Consider thesurface M ⊆ R3 given by the graph of f in R3. Let p ∈ R2 be a criticalpoint of f . Then the value of det(Hf(p)) is precisely the Gaussiancurvature of the surface M at the point (p, f(p)) ∈ R3.

See Definition 9.6.1 for more details. We will return to surfaces inSection 5.3.

3.8. Parametric representation of a curve

There are two main ways of representing a curve in the plane: para-metric and implicit.

A curve in the plane can be represented by a pair of coordinatesevolving as a function of the time parameter t:

α(t) = (α1(t), α2(t)), t ∈ [a, b],

with coordinates α1(t) and α2(t). Both functions are assumed to be ofclass C2. Thus a parametrized curve can be viewed as a map

α : [a, b] → R2. (3.8.1)

Let C be the image of the map (3.8.1). Then C ⊆ R2 is the geometriccurve independent of parametrisation. Thus, changing the parametri-sation by setting t = t(s) and replacing α by a new curve β(s) = α(t(s))preserves the geometric curve.

Definition 3.8.1. A parametrisation α(t) is called regular if forall t one has α′(t) 6= 0.

3.10. IMPLICIT FUNCTION THEOREM 39

3.9. Implicit representation of a curve

A curve in the (x, y)-plane can also be represented implicitly as thesolution set of an equation

F (x, y) = 0,

where F is a function always assumed to be of class C2.

Definition 3.9.1. The level curve CF ⊆ R2 is the locus of theequation

CF = {(x, y) : F (x, y) = 0}.Example 3.9.2. A circle of radius r > 0 corresponds to the choice

of the function F (x, y) = x2 + y2 − r2.

Further examples are given below.

(1) The function F (x, y) = y − x2 defines a parabola.(2) The function F (x, y) = xy − 1 defines a hyperbola.(3) The function F (x, y) = x2 − y2 − 1 also defines a hyperbola.

Remark 3.9.3. In each of these cases, it is easy to find a parametri-sation (at least of a part of) the level curve, by solving the equationfor one of the variables. Thus, in the case of the circle, we choose thepositive square root to obtain y =

√r2 − x2, giving a parametrisation

of the upperhalf circle by means of the pair of formulas

α1(t) = t, α2(t) =√r2 − t2.

Note this is not all of the level curve CF .

Unlike the above examples, typically it is difficult to find an explicitparametrisation for a curve defined by an implicit equation F (x, y) = 0.Locally one can always find one in theory under a suitable nondegener-acy condition, expressed by the implicit function theorem, dealt within Section 3.10.

3.10. Implicit function theorem

Theorem 3.10.1 (implicit function theorem). Assume the func-tion F (x, y) is C2. Suppose the gradient of F does not vanish at aspecific point p = (x, y) ∈ CF , in other words

∇F (p) 6= 0.

Then there exists a regular parametrisation (α1(t), α2(t)) of the levelcurve CF in a suitable neighborhood of p.

A useful special case is the following result.


Theorem 3.10.2 (implicit function theorem: special case). Assumethe function F (x, y) is C2. Suppose that the partial derivative withrespect to y satisfies

∂F

∂y(p) 6= 0.

Then there exists a parametrisation y = α2(x), in other words α(t) =(t, α2(t)) of the curve CF in a suitable neighborhood of p.

Example 3.10.3. In the case of the circle x2+y2 = r2, the point (r, 0)on the x-axis fails to satisfy the hypothesis of Theorem 3.10.2. Thecurve cannot be represented by a differentiable function y = y(x) ina neighborhood of this point due to the problem of a vertical tangent.On the other hand, a parametrisation still exists, e.g., (r cos t, r sin t).

3.11. Unit speed parametrisation

Consider a parametrized curve α(t) = (α1(t), α2(t)) in the plane.Denote by C the underlying geometric curve, i.e., the image of α:

C ={(x, y) ∈ R2 : (∃t)

(x = α1(t), y = α2(t)

)}.

Definition 3.11.1. We say α = α(t) is a unit speed curve if∣∣dαdt

∣∣ =1, i.e.

(dα1

dt

)2+(dα2

dt

)2= 1 for all t ∈ [a, b].

Definition 3.11.2. The parameter of a unit speed parametrisation,usually denoted s, is called arclength.

Example 3.11.3. Let r > 0. Then the curve

α(s) =(r cos

s

r, r sin

s

r

)

is a unit speed parametrisation of the circle of radius r. Indeed, we

have∣∣dαds

∣∣ =√(

r 1r

(− sin s

r

))2+(r 1rcos s

r

)2=√sin2 s

r+ cos2 s

r= 1.

A regular curve always admits an arclength parametrisation; seeTheorem 5.1.

CHAPTER 4

Curvature, Laplacian, Bateman–Reiss

4.1. Geodesic curvature

We review the standard calculus topic of the curvature of a curve.

Remark 4.1.1. The notion of curvature of curves is indispensableto understanding the principal curvatures of a surface; see e.g., Theo-rem 9.8.1 and Theorem 10.1.5.

Our main interest will be in space curves, when one cannot in gen-eral assign a sign to the curvature. Therefore in the definition below wedo not concern ourselves with the sign of the curvature of plane curves,either. In this section, only local properties of curvature of curves willbe studied.

Remark 4.1.2. For oriented plane curves, there is a finer invariantcalled signed curvature; see Definition 11.1.4. A global result on thecurvature of plane curves may be found in Section 11.6.

Definition 4.1.3. The (geodesic) curvature function kα(s) ≥ 0 ofa unit speed curve α(s) is defined by setting

kα(s) =

∣∣∣∣d2α

ds2

∣∣∣∣. (4.1.1)

Theorem 4.1.4. The curvature kα(s) of the circle of radius r > 0is kα(s) =

1rat every point of the circle.

Proof. With the parametrisation given in Example 3.11.3, wehave

d2α

ds2=

(r1

r2

(− cos

s

r

), r

1

r2

(− sin

s

r

))

for all s, and so kα =∣∣1r

(− cos s

r,− sin s

r

)∣∣ = 1r. �

Note that in this case, the curvature is independent of the point,i.e. is a constant function of s.

In Section 11.2, we will give a formula for curvature with respectto an arbitrary parametrisation (not necessarily arclength).

41

42 4. CURVATURE, LAPLACIAN, BATEMAN–REISS

In Section 11.4, the curvature will be expressed in terms of theangle θ formed by the tangent vector (to the curve) with the positive x-axis.

4.2. Gauss map; shape operator for curves

Consider a regular curve C parametrized by α(s) where s is thearclength.

Definition 4.2.1. The unit tangent vector at a point α(s) of C isthe vector v(s) = α′(s), written in coordinates as v(s) = (x(s), y(s)).

Definition 4.2.2. The unit normal vector to the curve is the vec-tor n(s) = (y(s),−x(s)).

Remark 4.2.3. Since |n(s)| = 1 by definition, one can think of n(s)

as a map C → S1, where S1 ={(x, y) :

√x2 + y2 = 1

}is the unit

circle. This leads us to the notion of Gauss map for the curve C; seeSection 11.4.

Lemma 4.2.4. Let v(s) = (x(s), y(s)) where v(s) is the unit tangentvector of α(s). Then

dy

ds= −dx

ds

x

y. (4.2.1)

Proof. We start with v(s) = (x(s), y(s)). Differentiating theidentity x(s)2 + y(s)2 = 1 with respect to s using chain rule, we ob-tain xdx

ds+ y dy

ds= 0 and therefore dy

ds= −dx

dsxyas required. �

Lemma 4.2.5. Let v(s) = (x(s), y(s)) where v(s) is the unit tangentvector of α(s). With respect to the arclength parameter s, we have thefollowing formula for the curvature:

kα(s) =

∣∣∣∣1

y

dx

ds

∣∣∣∣ .

4.2. GAUSS MAP; SHAPE OPERATOR FOR CURVES 43

Proof. We use Lemma 4.2.4. Substituting (4.2.1) into α′′ we ob-tain

α′′(s) =

(dx

ds,dy

ds

)

=

(dx

ds,−dx

ds

x

y

)

=dx

ds

(1,−x

y

)

=1

y

dx

ds(y,−x)

=

(1

y

dx

ds

)n

and therefore kα(s) = |α′′(s)| =∣∣∣ 1y dxds

∣∣∣, as required. �

Proposition 4.2.6. Differentiating the normal vector n(s) alongthe curve gives a vector proportional to v(s) with coefficient of propor-tionality up to sign given by the geodesic curvature kα(s):

d

dsn(s) = ±kα(s)v(s).

Proof. Differentiating and applying (4.2.1), we obtain

−n′ (s) =

(−dyds,dx

ds

)

=

(x

y

dx

ds,dx

ds

)

=dx

ds

(x

y, 1

)

=1

y

dx

ds(x, y)

=

(1

y

dx

ds

)v

and we apply Lemma 4.2.5 to prove the proposition. �

Remark 4.2.7. A similar phenomenon occurs for surfaces wherethe directional derivative of the normal vector n to the surface is usedto define the Weingarten map (shape operator) and ultimately theGaussian curvature K of the surface; see Section 8.10.


4.3. Osculating circle of a curve

To give a more geometric description of the curvature of a curve, wewill exploit its osculating circle1 at a point. We first recall the followingfact about the second derivative.

Theorem 4.3.1. Let f ∈ C2(I). The second derivative of a func-tion f(x) may be computed from a triple of points f(x), f(x+h), f(x−h)that are infinitely close to each other, as follows:

f ′′(x) = limh→0

f(x+ h) + f(x− h)− 2f(x)

h2.

The theorem remains valid for vector-valued functions; see Defini-tion 4.3.2. We see that the second derivative can be calculated fromthe value of the function at a triple of nearby points x, x+ h, x− h.

Definition 4.3.2. The osculating circle to the curve parametrizedby α with arclength parameter s at the point α(s) is obtained by choos-ing a circle passing through the three points

α(s), α(s− h), α(s+ h)

for infinitesimal h. The standard part2 of the resulting circle is theosculating circle; equivalently, we take limit as h tends to zero.

Remark 4.3.3. The osculating circle and the curve are “better thantangent” (they have second order tangency).

Theorem 4.3.4. The curvatures of the osculating circle and thecurve at the point of tangency are equal.3

Proof. The curvature is defined by the second derivative. Thesecond derivative is computed from the same triple of points for α(s)and for the osculating circle (cf. Remark 4.3.1), proving the theorem(cf. [We55, p. 13]). �

1Maagal noshek2See Keisler [Ke74].3For example, let y = f(x). Compute the curvature of the graph of f

when f(x) = ax2. Let B = (x, x2). Let A be the midpoint of OB. Let C be theintersection of the perpendicular bisector of OB with the y-axis. Let D = (0, x).

Triangle OAC yields sinψ = OAOC

=12

√x2+(ax2)2

r, triangle OBD yields sinψ =

BDOB

= ax2√x2+a2x4

, and12

√x2+a2x4

r= ax2

√x2+a2x4

, so that 12 (x

2 + a2x4) = arx2,

and 12 (1 + a2x2) = ar, so that r = 1+a2x2

2a . Taking the limit as x→ 0, we ob-

tain r = 12a , hence k = 1

r= 2a = f ′′(0). Thus the curvature of the parabola at

its vertex equals the second derivative with respect to x (even though x is not thearclength parameter of the graph).

4.5. FLAT LAPLACIAN 45

4.4. Center of curvature, radius of curvature

Recall that we denote the geometric curve by C ⊆ R2 and itsparametrisation by α(s) where s is the arclength parameter.

To help geometric intuition, it is useful to recall Leibniz’s andCauchy’s definition of the radius of curvature of a curve [Cauchy 1826]as the distance from the curve to the intersection point of two infinitelyclose normals to the curve. In more detail, we have the following.

Definition 4.4.1. Let p ∈ C. The center of curvature is the in-tersection point of the normals to the curve at infinitely close points pand p′ of C.

Theorem 4.4.2 (Relation between center of curvature and oscu-lating circle). Let p ∈ C. The center of curvature, i.e., the intersectionpoint of two infinitely close normals, is the center of the osculatingcircle to the curve at the point p ∈ C.

Definition 4.4.3. Let p ∈ C. The radius of curvature R of acurve C at p is the distance from p to the center of curvature. Thusthe curvature kα is the inverse of the radius of curvature: kα = 1

R.

4.5. Flat Laplacian

Computing the curvature of implicitly defined curves will involve acertain second-order differential operator. We start with the simplestexample of a second-order operator, the Laplacian.

Definition 4.5.1. The flat Laplacian ∆0 on R2 is the differentialoperator is defined by

∆0 =∂2

∂x2+

∂2

∂y2.

Remark 4.5.2. The formula means that when we apply ∆ to asmooth function F = F (x, y), we obtain

∆0F =∂2F

∂x2+∂2F

∂y2.

We also introduce the traditional shorter notation

Fxx =∂2F

∂x2, Fyy =

∂2F

∂y2, Fxy =

∂2F

∂x ∂y.

Then we obtain the equivalent formula

∆0(F ) = Fxx + Fyy.

Example 4.5.3. If F (x, y) = x2 + y2 − r2, then ∂2F∂x2

= 2 and

similarly ∂2F∂y2

= 2, hence ∆0F = 4.


4.6. Bateman–Reiss operator

Recall that we have two types of presentations of a curve, eitherparametric or implicit.

In Section 4.1 the curvature of a curve was calculated starting witha parametric representation of the curve. If a curve is given implicitlyas the locus (solution set) of an equation F (x, y) = 0, one can alsocalculate the geodesic curvature, by means of the theorems given inthe Section 4.7. We start with a definition.

We will be interested in the following operator that appears in aformula for curvature.

Definition 4.6.1. The Bateman-Reiss operator DB is defined by

DB(F ) = FxxF2y − 2FxyFxFy + FyyF

2x , (4.6.1)

which is a non-linear second order differential operator.

The subscript “B” stands for Bateman, as in the Bateman equa-tion FxxF

2y − 2FxyFxFy + FyyF

2x = 0.

Theorem 4.6.2. The Bateman operator can be represented by thedeterminant

DB(F ) = −det

0 Fx FyFx Fxx FxyFy Fxy Fyy

. (4.6.2)

Proof. Expanding the determinant (4.6.2) along the first row, weobtain the formula (4.6.1). �

This operator was treated in detail in [Goldman 2005, p. 637,formula (3.1)]. The same operator occurs in the Reiss relation in alge-braic geometry; see Griffiths and Harris [GriH78, p. 677].4 See alsoValenti [2, p. 804] who refers to geodesic curvature as isophote curva-ture in the context of a study of luminosity and eye center location.See also [3] and (two) references therein.

4.7. Geodesic curvature for an implicit curve

The operator defined in Section 4.6 allows us to calculate the cur-vature of a curve presented in implicit form, without having to spec-ify a parametrisation. Let CF ⊆ R2 be a curve defined implicitlyby F (x, y) = 0.

4Michel Reiss (1805-1869).

4.8. CURVATURE OF CONIC SECTIONS 47

Theorem 4.7.1. Let p ∈ CF , and suppose ∇F (p) 6= 0. Then thegeodesic curvature kC of CF at the point p is given by

kC =|DB(F )||∇F |3 ,

where DB is the Bateman–Reiss operator defined in (4.6.1).

A proof can be found in [Goldman 2005].

4.8. Curvature of conic sections

Using the formula for curvature based on the Bateman–Reiss oper-ator, one can calculate the curvature and solve maximization problems.

Example 4.8.1. The circle C of radius r is defined by the equa-tion F (x, y) = 0, where F = x2 + y2 − r2. Then

Fx = 2x, Fy = 2y, ∇F = (2x, 2y)t,

and therefore |∇F | = 2√x2 + y2 = 2r. Meanwhile, Fxx = 2, Fyy =

2, Fxy = 0, hence

DB(F ) = 2(2y)2 + 2(2x)2 = 8r2,

and therefore the curvature is kC = 8r2

8r3= 1

r, in accordance with Theo-

rem 4.1.4.

Example 4.8.2. Consider the parabola C = CF given by the zerolocus of F (x, y) = y − x2. Then Fx = −2x, Fy = 1, and |∇F | =√1 + 4x2. Meanwhile, Fxx = −2, Fyy = 0, Fxy = 0. Applying the

formula DB(F ) = FxxF2y − 2FxyFxFy + FyyF

2x we obtain DB(F ) =

−2(1)2 = −2.Thus the curvature satisfies kC = 2

(1+x2)3/2allowing us to identify

the point of maximal curvature in Theorem 4.8.3 as follows.

Theorem 4.8.3. The apex of the parabola y = x2 is its point ofmaximal curvature.

Proof. The calculations above produce kC = 2(1+x2)3/2

. Since 1 +

x2 ≥ 1, the curvature attains its maximal value k = 2 when x = 0. �

Example 4.8.4. Consider the hyperbola C = CF given by the zerolocus of F (x, y) = xy−1. Then Fx = y, Fy = x, and |∇F | =

√x2 + y2.

Meanwhile Fxx = 0, Fyy = 0, and Fxy = 1. Hence DB(F ) = −2xy. Tosimplify the expression for DB(F ) we exploit the defining equation ofthe curve xy = 1. Hence we have DB(F ) = −2 at every point of thehyperbola.


Hence kC = 2(x2+y2)3/2

, enabling us to find the maximum in Theo-

rem 4.8.5 as follows.

Theorem 4.8.5. The curvature of the hyperbola C defined by theequation xy = 1 is maximal at the point (1, 1).

Proof. The calculations in Example 4.8.4 produce kC = 2(x2+y2)3/2

.

To maximize this expression along the curve, it suffices to minimize theexpression x2+ y2 whose power appears in the denominator, restrictedto the curve itself. We exploit the defining relation xy = 1 of the curveto obtain

x2 + y2 = (x+ y)2 − 2xy = (x+ y)2 − 1 =(x+ 1

x

)2 − 1.

To minimize this expression, it suffices to minimize the quantity x+ 1x.

The sum x+ 1xis known to be minimal when x = 1. Namely, to show

that x+ 1x≥ 2, rewrite the inequality as x2+1 ≥ 2x or x2+1−2x ≥ 0

or equivalently (x − 1)2 ≥ 0 which is a true inequality. Hence themaximum of the curvature is when x = 1 and so y = 1, proving thetheorem.

The same result can be obtained by differentiating the expression(x+ 1

x

)2 − 1. �

4.9. Maximal curvature of a transcendental curve

Theorem 4.9.1. Consider the logarithmic curve y = ln x, x > 0.Then the maximum of curvature of the curve is attained at the point x =1√2, y = −1

2ln 2.

Proof. Let F (x, y) = y−ln x. Then Fx = − 1x, Fy = 1, and |∇F | =√

1 + x−2. Then Fxx = 1x2, Fyy = 0, and Fxy = 0. Applying the

formula DB(F ) = FxxF2y − 2FxyFxFy + FyyF

2x we obtain DB(F ) =

1x2

· 1− 2 · 0 + 0 = 1x2

= x−2. The curvature of the logarithmic curve istherefore

k =x−2

(√1 + x−2)3

=1

x2(1 + x−2)3/2=

x3

x2(x2 + 1)3/2=

x

(x2 + 1)3/2.

To maximize k it is sufficient to maximize k2 = x2

(x2+1)3.

We will use an auxiliary variable z = x2 to simplify calculations.We are therefore interested in maximizing the expression g(z) = z

(z+1)3.

4.10. EXPLOITING THE DEFINING EQUATION IN STUDYING CURVATURE49

Differentiating we obtain

g′(z) =(z + 1)3 · 1− z · 3(z + 1)2

(z + 1)6

=z + 1− 3z

(z + 1)4=

1− 2z

(z + 1)4= 0.

We obtain an extremum when 1 − 2z = 0, i.e., z = 12. Checking

that the second derivative is negative at the point, we conclude thatthis is a point of maximum. Thus the maximum of the curvatureof the logarithmic curve is attained when x2 = 1

2, i.e., x = 2−1/2.

Hence y = ln(2−1/2) = −12ln 2. �

4.10. Exploiting the defining equation in studying curvature

The defining equation of the curve may be exploited several timesin the course of the calculation when studying the curvature of a curve.

Theorem 4.10.1. The maxima of the curvature of the curve defined

by the equation xy + y2 = 1 are given by x = 1−√2

± 4√2and y = ± 4

√2.

Proof. Set F (x, y) = xy+ y2− 1. Then Fx = y, Fy = x+2y, and

|∇F | =√y2 + (x+ 2y)2 =

√y2 + x2 + 4xy + 4y2

=√y2 + x2 + 4(xy + y2).

4.10.1. Step 1. We use the defining equation of the curve to re-place xy + y2 by 1, obtaining

|∇F | =√x2 + y2 + 4. (4.10.1)

Next, Fxx = 0, Fxy = 1, and Fyy = 2. Therefore

DB(F ) = 0− 2xy − 4y2 + 2y2 = −2(xy + y2). (4.10.2)

4.10.2. Step 2. Exploiting the defining relation of the curve, weobtain from equation (4.10.2) that

DB(F ) = −2. (4.10.3)

Using (4.10.1) and (4.10.3), we obtain that the curvature at a point (x, y)of the curve is

k =|DB(F )||∇F |3 =

2

(x2 + y2 + 4)3/2.

The curvature along the curve is maximal when the expression x2+y2+4 along the curve is minimal, or equivalently when x2 + y2 is minimalalong the curve.


4.10.3. Step 3. Using the defining relation of the curve we obtain

x =1− y2

y=

1

y− y. (4.10.4)

Therefore x2 + y2 = (1−y2)2y2

+ y2 = (1−y2)2+y4y2

. To simplify calculations,

we introduce an auxiliary variable z = y2. Then the expression to beminimized is

g(z) =(1− z)2 + z2

z=

2z2 − 2z + 1

z= 2z +

1

z− 2.

We set g′(z) = 2− 1z2

= 0 to find the extremum z =√2. Checking that

the second derivative is positive, we conclude that y = ±√z = ± 21/4

and the corresponding x can be found from x = 1−y2y

. This gives the

maximum of the curvature for the curve (4.10.4). �

4.11. Curvature of graph of function

Theorem 4.11.1. Let c be a critical point of f(x), and considerthe graph of f at the point (c, f(c)). Then the curvature k of the graphequals

k = |f ′′(c)|.Proof. We parametrize the graph by α(t) = (t, f(t)). Then we

have α′′(t) = (0, f ′′(t)) and |α′′(t)| = |f ′′(t)|, which is the expectedanswer at a critical point. This is merely a heuristic calculation becausethe parametrisation (t, f(t)) is not a unit speed parametrisation of thegraph.

We apply the characterisation of curvature in terms of the BatemanoperatorDB(F ) = FxxF

2y−2FxyFxFy+FyyF

2x . Let F (x, y) = −f(x)+y.

At the critical point c, we have

∇F = (−f ′(c), 1)t = (1, 0)t,

whileDB(F ) = −f ′′(x)(1)2 − 0 + 0 = −f ′′(x).

Hence the curvature at this point satisfies

k =|DBF ||∇F |3 =

|f ′′(c)|1

= |f ′′(c)|,

as required. �

CHAPTER 5

Surfaces and their curvature

5.1. Existence of arclength

The unit speed parametrisation i.e., the arclength parametrisationof the geometric curve C exists for all regular curves.

Theorem 5.1.1. Let α(t) be a regular parametrisation of the geo-metric curve C. Then there exists a unit speed parametrition β(s) =α(t(s)) of the curve C, where the parameter s is defined by s(t) =∫ t

a

∣∣∣dαdτ

∣∣∣dτ .

Proof. Recall the formula for the length of the graph of f(x)from a to b:

L =

∫ b

a

√1 + (f ′(x))2 dx .

The element of length (infinitesimal increment) ds along the graphdecomposes by Pythagoras’ theorem as follows:

ds2 = dx2 + dy2.

More generally, for a curve α(t) = (α1(t), α2(t)), we have the followingformula for the length:

L =

∫ b

a

ds

=

∫ b

a

√dx2 + dy2

=

∫ b

a

√(d(α1)

dt

)2+(dα2

dt

)2dt

=

∫ b

a

∣∣∣dαdt

∣∣∣dt.

We define the new parameter s = s(t) by setting

s(t) =

∫ t

a

∣∣∣dαdτ

∣∣∣dτ, (5.1.1)

51

52 5. SURFACES AND THEIR CURVATURE

where τ is a dummy variable (internal variable of integration). By the

Fundamental Theorem of Calculus, we haveds

dt=∣∣∣dαdt

∣∣∣. The function

being monotone increasing, there exists an inverse function t = t(s).Let β(s) = α(t(s)). Then by chain rule

dβ

ds=dα

dt

dt

ds=dα

dt

1

ds/dt=dα

dt

1

|dα/dt| .

Thus∣∣dβds

∣∣ = 1. �

5.2. Non-existence of regular parametrisation

Example 5.2.1 (Cusp). The plane curve α(t) = (t3, t2) is smoothbut not regular. Its graph exhibits a cusp. In this case it is impossibleto find an arclength parametrisation of the curve.

Remark 5.2.2 (Curves in R3). A space curve may be written incoordinates as

α(s) = (α1(s), α2(s), α3(s)).

Here s is the arc length if∣∣∣dαds∣∣∣ = 1 i.e.

∑3i=1

(dαi

ds

)2= 1.

Example 5.2.3. Helix α(t) = (a cosωt, a sinωt, bt).(i) make a drawing in case a = b = ω = 1.(ii) parametrize by arc length.(iii) compute the curvature.

5.3. Arnold’s observation on folding a page

The differential geometry of surfaces in Euclidean 3-space startswith the observation that they inherit a metric structure from the am-bient space (i.e. the Euclidean space).

Question 5.3.1. We would like to understand which geometricproperties of this structure are intrinsic?1

Part of the job is to clarify the sense of the term intrinsic.

Remark 5.3.2. Following Arnold [Ar74, Appendix 1, p. 301], notethat a piece of paper may be placed flat on a table, or it may be rolledinto a cylinder, or it may be rolled into a cone. However, it cannotbe transformed into the surface of a sphere, that is, without tearing orstretching. Understanding this phenomenon quantitatively is our goal,cf. Figure 12.2.1.

1This is atzmit rather than pnimit according to Vishne.

5.4. REGULAR SURFACE; JACOBIAN 53

5.4. Regular surface; Jacobian

Our starting point is the first fundamental form, obtained by re-stricting the 3-dimensional inner product. What does the first funda-mental form measure? A helpful observation to keep in mind is that itenables one to measure the length of curves on the surface.

Consider a surface M ⊆ R3 parametrized by a map x(u1, u2) or

x : R2 → R3. (5.4.1)

We will always assume that x is differentiable. From now on we willfrequently omit the underline in x, and write simply “x”, to lighten thenotation.

Definition 5.4.1. The Jacobian matrix Jx is the 3× 2 matrix

Jx =

(∂xi

∂uj

), (5.4.2)

where xi are the three components of the vector valued function x.

Definition 5.4.2. The parametrisation x of the surfaceM is calledregular if one of the following equivalent conditions is satisfied:

(1) the vectors ∂x∂u1

and ∂x∂u2

are linearly independent;

(2) the vector product x1 × x2 is nonzero, where xi =∂x∂ui

;(3) the Jacobian matrix (5.4.2) is of rank 2.

Example 5.4.3. Let b > 0 be a fixed real number, and consider thefunction f(x, y) =

√b2 − x2 − y2. The graph of f can be parametrized

as follows:x(u1, u2) = (u1, u2, f(u1, u2)).

This provides a parametrisation of the (open) northern hemisphere.

Then ∂x∂u1

= (1, 0, fx(u1, u2))t while ∂x

∂u2= (0, 1, fy(u

1, u2))t. These vec-tors are linearly independent and therefore we have a regular parametri-sation.

Note that we have the relations

fx =−xf

=−x√

b2 − x2 − y2, fy =

−yf. (5.4.3)

The southern hemisphere can be similarly parametrized by

x′ = (u1, u2,−f(u1, u2)).The formulas for the ∂x′

∂uiare then ∂x′

∂u1= (1, 0,−fx)t = (1, 0, x√

b2−x2−y2),

while ∂x′

∂u2= (0, 1,−fy)t = (0, 1, y√

b2−x2−y2).


5.5. Coefficients of first fundamental form of a surface

Let 〈 , 〉 denote the inner product in R3. For i = 1, 2 and j = 1, 2,define functions gij = gij(u

1, u2) called “metric coefficients” by

gij(u1, u2) =

⟨ ∂x∂ui

,∂x

∂uj

⟩. (5.5.1)

Remark 5.5.1. We have gij = gji as the inner product is symmet-ric.

Example 5.5.2 (Metric coefficients for the graph of a function).The graph of a function f satisfies

⟨∂x

∂u1,∂x

∂u1

⟩= 1 + f 2

x ,

⟨∂x

∂u1,∂x

∂u2

⟩= fxfy,

⟨∂x

∂u2,∂x

∂u2

⟩= 1 + f 2

y .

Therefore in this case we have

(gij) =

(1 + f 2

x fxfyfxfy 1 + f 2

y

)

In the case of the hemisphere, the partial derivatives are as in for-mula (5.4.3).

5.6. Metric coefficients in spherical coordinates

Spherical coordinates2 (ρ, θ, ϕ) in 3-space will be reviewed in moredetail in Section 6.8. Briefly, ϕ is the angle formed by the positionvector of the point with the positive direction of the z-axis. Mean-while, θ is the polar coordinate angle θ for the projection of the vectorto the x, y plane.

Example 5.6.1 (Metric coefficients in spherical coordinates). Con-sider the parametrisation of the unit sphere S2 by means of sphericalcoordinates, so that x = x(θ, ϕ). We have

x = sinϕ cos θ, y = sinϕ sin θ, z = cosϕ .

Setting u1 = θ and u2 = ϕ, we obtain

∂x

∂u1= (− sinϕ sin θ, sinϕ cos θ, 0)

2Koordinatot kaduriot

5.8. FIRST FUNDAMENTAL FORM OF M 55

and∂x

∂u2= (cosϕ cos θ, cosϕ sin θ,− sinϕ),

Thus in this case we have

(gij) =

(sin2 ϕ 00 1

)

so that det(gij) = sin2 ϕ and√det(gij) = sinϕ. This expression will

appear in the formula for the area.

5.7. Tangent plane

Consider a parametrisation x(u1, u2) of a surface M ⊆ R3 andlet p = x(u1, u2) ∈ R3 be a point on the surface.

Definition 5.7.1. The vectors xi =∂x

∂ui, i = 1, 2, are called the

tangent vectors to the surface M at the point p = x(u1, u2).

Definition 5.7.2. Given an ordered n-tuple S = {vi}i=1,...,n in Rb,we define its Gram matrix as the matrix of inner products

Gram(S) = (〈vi, vj〉)i=1,...,n; j=1,...,n. (5.7.1)

In Section 5.6 we defined the metric coefficients gij = gij(u1, u2).

We now state a relationship between the Gram matrix and the metriccoefficients.

Theorem 5.7.3. The 2× 2 matrix (gij) is the Gram matrix of thepair of tangent vectors:

(gij) = JTx Jx,

cf. formula (15.6.1).

The proof is immediate.

Definition 5.7.4. The plane in R3 spanned by the vectors x1(u1, u2)

and x2(u1, u2) is called the tangent plane to the surface M at the

point p = x(u1, u2), and denoted

TpM.

5.8. First fundamental form of M

The tangent plane TpM of a surface M ⊆ R3 was defined in Sec-tion 5.7.


Definition 5.8.1. The first fundamental form Ip of the surface Mat the point p is the bilinear form on the tangent plane Tp, namely

Ip : TpM × TpM → R,

defined by the restriction of the ambient Euclidean inner product:

Ip(v, w) = 〈v, w〉R3,

for all v, w ∈ TpM . With respect to the basis (frame) (x1, x2), it isgiven by the matrix (gij), where

gij = 〈xi, xj〉.The coefficients gij are sometimes called ‘metric coefficients.’

5.9. Ip of plane and cylinder

Like curves, surfaces can be represented either implicitly or para-metrically (see Section 3.8 for representations of curves).

The example of the sphere was discussed in Section 5.6. We nowconsider two more examples.

Example 5.9.1 (Plane). The x, y-plane in R3 is defined implic-itly by the equation z = 0. Consider the parametrisation x(u1, u2) =(u1, u2, 0) ∈ R3. This is a parametrisation of the xy-plane in R3.Then x1 = (1, 0, 0)t, x2 = (0, 1, 0)t, and

g11 = 〈x1, x1〉 =⟨100

,

100

⟩

= 1,

etcetera. Thus we have (gij) =

(1 00 1

)= I.

Example 5.9.2 (Cylinder). Let x(u1, u2) = (cosu1, sin u1, u2). Thisformula provides a parametrisation of the cylinder. We have

x1 = (− sin u1, cosu1, 0)t

and x2 = (0, 0, 1)t, while

g11 =

⟨− sin u1

cosu1

0

,

− sin u1

cosu1

0

⟩

= sin2 u1 + cos2 u1 = 1,

etc. Thus (gij) =

(1 00 1

)= I.

5.10. SURFACES OF REVOLUTION 57

Remark 5.9.3. The two examples above illustrate that the firstfundamental form does not contain all the information (even up toEuclidean congruences) about the surface. Indeed, the plane and thecylinder have the same first fundamental form, but are geometricallydistinct embedded surfaces.

5.10. Surfaces of revolution

Definition 5.10.1 (Surfaces of revolution). Here it is customary touse the notation u1 = θ and u2 = φ. The starting point is a generatingcurve C in the xz-plane, parametrized by a pair of functions

x = r(φ), z = z(φ).

The surface of revolution (around the z-axis) generated by C is para-metrized as follows:

x(θ, φ) = (r(φ) cos θ, r(φ) sin θ, z(φ)). (5.10.1)

Example 5.10.2. Consider a generating curve which is the verticalline r(φ) = 1, z(φ) = φ. The resulting surface of revolution is thecylinder.

Example 5.10.3. The generating curve r(φ) = sinφ, z(φ) = cos φyields the sphere S2 in spherical coordinates as discussed in Section 5.6;see Example 5.10.5 for more details.

Theorem 5.10.4. If φ is the arclength parameter of a parametrisa-tion (r(φ), z(φ)) of the generating curve C, then the first fundamentalform of the corresponding surface of revolution (5.10.1) is given by

(gij) =

(r2(φ) 00 1

).

Proof. We have

x1 =∂x

∂θ= (−r sin θ, r cos θ, 0)t,

x2 =∂x

∂φ=

(dr

dφcos θ,

dr

dφsin θ,

dz

dφ

)t

therefore

g11 =

∣∣∣∣∣∣∣

−r sin θr cos θ

0

∣∣∣∣∣∣∣

2

= r2 sin2 θ + r2 cos2 θ = r2


and

g22 =

∣∣∣∣∣∣

drdφ

cos θdrdφ

sin θdzdφ

∣∣∣∣∣∣

2

=

(dr

dφ

)2

(cos2 θ + sin2 θ) +

(dz

dφ

)2

=

(dr

dφ

)2

+

(dz

dφ

)2

and

g12 =

⟨−r sin θr cos θ

0

,

drdφ

cos θdrdφ

sin θdzdφ

⟩

= −r drdφ

sin θ cos θ + rdr

dφcos θ sin θ = 0.

Thus

(gij) =

(r2 0

0(drdφ

)2+(dzdφ

)2).

In the case of an arclength parametrisation of the generating curve C,we obtain g22 = 1, proving the theorem. �

Example 5.10.5. Consider the curve (sin φ, cosφ) in the x, z-plane.The resulting surface of revolution is the sphere, where the φ parame-ter coincides with the angle ϕ of spherical coordinates. Thus, for thesphere S2 we obtain

(gij) =

(sin2 φ 00 1

).

5.11. From tractrix to pseudosphere

Example 5.11.1. The pseudosphere (so called because its Gauss-ian curvature equals −1) is the surface of revolution generated by acurve called the tractrix. This curve is parametrized by (r(φ), z(φ))where r(φ) = eφ and

z(φ) =

∫ φ

0

√1− e2ψdψ = −

∫ 0

φ

√1− e2ψdψ,

where −∞ < φ ≤ 0. The tractrix generates a surface of revolutionwith g11 = e2φ, while

g22 = (eφ)2 + (√

1− e2φ)2

= e2φ + 1− e2φ = 1.

5.12. CHAIN RULE IN TWO VARIABLES 59

Thus (gij) =

(e2φ 00 1

).

Its Gaussian curvature is calculated in Section 13.7.

5.12. Chain rule in two variables

How does one measure the length of curves on a surface in terms ofthe metric coefficients of the surface x(u1, u2)? Let

α : [a, b] → R2, α(t) = (α1(t), α2(t))

be a plane curve. We will exploit the Einstein summation convention.We will also use the following version of chain rule in several variables.

Theorem 5.12.1 (Chain rule in two variables). Consider the curveon the surface defined by the composition β(t) = x ◦ α(t) where x =

x(u1, u2). Then dβdt

=∑2

i=1∂x∂ui

dαi

dt, or in Einstein summation conven-

tiondβ

dt=

∂x

∂uidαi

dt= Jx

(dαdt

)t

(the superscript t is the transpose).

CHAPTER 6

Γ symbols

6.1. Measuring length of curves on surfaces

In Section 5.6, we defined the metric coefficients gij = 〈xi, xj〉 of asurface M with parametrisation x(u1, u2). The first fundamental form

Ip : TpM × TpM → R,

represented by the matrix (gij), determines the length of curves on thesurface as follows.

Theorem 6.1.1. Let β(t) = x ◦ α(t), t ∈ [a, b], be a curve on thesurface with parametrisation x. Then the length L of β is given by theformula

L =

∫ b

a

√gij(α(t))

dαi

dt

dαj

dtdt.

Proof. The length L of β is calculated as follows using chain rule:

L =

∫ b

a

∥∥∥∥dβ

dt

∥∥∥∥ dt

=

∫ b

a

∥∥∥∥∥

2∑

i=1

∂x

∂uidαi

dt

∥∥∥∥∥ dt

=

∫ b

a

√⟨xidαi

dt, xj

dαj

dt

⟩dt

=

∫ b

a

√〈xi, xj〉

dαi

dt

dαj

dtdt

=

∫ b

a

√gij(α(t))

dαi

dt

dαj

dtdt.

We therefore obtain L =∫ ba

√gij(α(t))

dαi

dtdαj

dtdt as required. �

Corollary 6.1.2. If (gij) = (δij) is the identity matrix, then thelength of the curve β = x ◦ α on the surface equals the length of thecurve α of R2.

61

62 6. Γ SYMBOLS

Proof. If the first fundamental form of the surface satisfies gij =

δij, then L =

∫ b

a

√(dα1

dt

)2

+

(dα2

dt

)2

dt, namely the length of the

curve α(t) in R2. �

Remark 6.1.3. To simplify notation, let x = x(t) = α1(t) andy = y(t) = α2(t). Then the dt cancels out. The infinitesimal element

of arclength is ds =√dx2 + dy2. The length of the curve is

∫ds =∫ √

dx2 + dy2.

6.2. Normal vector to a surface

Let x : R2 → R3 be a regular parametrized surface, and let xi =∂x∂ui

.

Definition 6.2.1. The normal vector n(u1, u2) to a regular surfaceat the point x(u1, u2) ∈ R3 is defined in terms of the vector product,cf. Definition 1.9.7, as follows:

n =x1 × x2|x1 × x2|

,

so that 〈n, xi〉 = 0 ∀i.Remark 6.2.2. Either the vector x1×x2

|x1×x2| or the vector

x2 × x1|x1 × x2|

= − x1 × x2|x1 × x2|

can be taken to be a normal vector to the surface.

Theorem 6.2.3. Consider the surface M ⊆ R3 given by the graph

of a function f(x, y). Then the vector

fxfy−1

is perpendicular to the

tangent plane Tp to M at the point p = (x, y, f(x, y)).

Proof. A standard parametrisation of the graph of f is x(u1, u2) =(u1, u2, f(u1, u2)). Then x1 = (1, 0, fx)

t and x2 = (0, 1, fy)t. Taking the

cross product, we obtain

det

−→i

−→j

−→k

1 0 fx0 1 fy

= −fx

−→i − fy

−→j +

−→k

which is the opposite of the vector

fxfy−1

. �

6.4. BASIC PROPERTIES OF THE Γ SYMBOLS 63

Corollary 6.2.4. The normal vector n of the graph of f(x, y) is

n =(fx, fy,−1)t√f 2x + f 2

y + 1.

Proof. This is immediate from the theorem by normalizing thevector (fx, fy,−1)t. �

6.3. Γ symbols of a surface

The symbols Γkij, roughly speaking,1 account for how the surface

twists2 in space. They are, however, coordinate dependent.

Remark 6.3.1. We will see that the symbols Γkij also control thebehavior of geodesics on the surface. Here geodesics on a surface canbe thought of as curves that are to the surface what straight lines areto a plane, or what great circles are to a sphere; cf. Definition 7.8.2.

If x is regular then the vectors x1, x2, n form a basis (frame) for R3.Recall that the metric coefficients gij were defined in terms of first par-tial derivatives of x. Meanwhile, the symbols Γkij are defined in termsof the second partial derivatives (since they are not even tensors, onedoes not stagger the indices). Namely, they are defined as the coeffi-cients of the decomposition of the second partial derivative vector xij ,defined by

xij =∂2x

∂ui∂uj

with respect to the basis, or frame,3 (x1, x2, n), as follows.

Definition 6.3.2. The symbols Γkij are uniquely determined by theformula

xij = Γ1ijx1 + Γ2

ijx2 + Lijn

(the coefficients Lij will be discussed in Section 9.7).

6.4. Basic properties of the Γ symbols

Proposition 6.4.1. We have the following formula for the Gammasymbols:

Γkij = 〈xij , xℓ〉gℓk,where (gℓk) is the inverse matrix of (gij).

1In the first approximation: kiruv rishon.2Mitpatelet3Maarechet yichus

64 6. Γ SYMBOLS

Proof. We have

〈xij , xℓ〉 = 〈Γkijxk + Lijn, xℓ〉= 〈Γkijxk, xℓ〉+ 〈Lijn, xℓ〉= Γkijgkℓ

since n is perpendicular to each tangent vector of the surface. We nowmultiply by gℓm and sum over ℓ:

〈xij , xℓ〉gℓm = Γkijgkℓgℓm = Γkijδ

mk = Γmij .

This is equivalent to the desired formula. �

Remark 6.4.2. We have the following relation: Γkij = Γkji, or Γk[ij] =

0 ∀ijk.Example 6.4.3 (The plane). Calculation of the symbols Γkij for

the plane x(u1, u2) = (u1, u2, 0). We have x1 = (1, 0, 0)t, x2 = (0, 1, 0)t.Thus we have xij = 0 ∀i, j. Hence Γkij = 0 for all i, j, k.

Example 6.4.4 (The cylinder). Calculation of the symbols for thecylinder x(u1, u2) = (cosu1, sin u1, u2). The normal vector is n =(cosu1, sin u1, 0)t, while x1 = (− sin u1, cosu1, 0)t, x2 = (0, 0, 1)t. Thuswe have x22 = 0, x21 = 0 and so Γk22 = 0 and Γk12 = Γk21 = 0 ∀k.

Meanwhile, x11 = (− cos u1,− sin u1, 0)t. This vector is propor-tional to n:

x11 = 0x1 + 0x2 + (−1)n.

Hence Γk11 = 0 ∀k.An example of nonzero Γ symbols will appear in Section 6.7.

6.5. Derivatives of the metric coefficients

Definition 6.5.1. We will use the following notation for the partialderivative of the function gij = gij(u

1, u2):

gij;k =∂

∂uk(gij).

Lemma 6.5.2. Scalar product of vector valued functions satisfiesLeibniz’s rule:

〈f, g〉′ = 〈f ′, g〉+ 〈f, g′〉. (6.5.1)

Proof. See [Leib]. In more detail, let (f1, f2) be components of f ,and let (g1, g2) be components of g. Then

〈f, g〉′ = (f1g1 + f2g2)′ = f1g

′1 + f ′

1g1 + f2g′2 + f ′

2g2 = 〈f ′, g〉+ 〈f, g′〉.The same proof goes through for arbitrary number of components, e.g.,for functions with values in R3. �

6.6. INTRINSIC NATURE OF THE Γ SYMBOLS 65

Lemma 6.5.3. In terms of the symmetrisation notation introducedin Section 1.6, we have

gij;k = 2gm{iΓmj}k. (6.5.2)

or more explicitly

gij;k = gmiΓmjk + gmjΓ

mik. (6.5.3)

Proof. Indeed, by Leibniz rule

gij;k =∂

∂uk〈xi, xj〉

= 〈xik, xj〉+ 〈xi, xjk〉= 〈Γmikxm, xj〉+ 〈Γmjkxm, xi〉.

By definition of the metric coefficients, we have

gij;k = Γmikgmj + Γmjkgmi

= gmjΓmik + gmiΓ

mjk

= 2gm{jΓmi}k

= 2gm{iΓmj}k

as required. �

Remark 6.5.4. The system of equations of type (6.5.3) suggeststhat one may be able to solve the system for Γkij , i.e., to express Γkijin terms of the metric coefficients gij and their derivatives gij;k. Thisturns out to be correct, as we show in Section 6.6.

6.6. Intrinsic nature of the Γ symbols

The coefficients Γ are intrinsic4 meaning that they are determinedby the metric coefficients alone, and are therefore independent of theambient (extrinsic) geometry of the surface, i.e., the way the surface“sits” in 3-space.

Theorem 6.6.1. The symbols Γkij can be expressed in terms of thefirst fundamental form and its derivatives as follows:

Γkij =1

2(giℓ;j − gij;ℓ + gjℓ;i)g

ℓk,

where gij is the inverse matrix of gij.

4This is atzmit and not pnimit according to Vishne.

66 6. Γ SYMBOLS

Γ1ij j = 1 j = 2

i = 1 0 1rdrdφ

i = 2 1rdrdφ

0

Table 6.7.1. Symbols Γ1ij of a surface of revolution (6.7.1)

Proof. The proof is a calculation. Applying Lemma 6.5.3 threetimes, we obtain

giℓ;j − gij;ℓ + gjℓ;i = 2gm{iΓmℓ}j − 2gm{iΓ

mj}ℓ + 2gm{jΓ

mℓ}i

= gmiΓmℓj + gmℓΓ

mij − gmiΓ

mjℓ − gmjΓ

miℓ + gmjΓ

mℓi + gmℓΓ

mji

= 2gmℓΓmji .

Thus1

2(giℓ;j−gij;ℓ+gjℓ;i)gℓk = Γmij gmℓg

ℓk = Γmij δkm = Γkij, as required. �

6.7. Γ Symbols for a surface of revolution

Recall that a surface of revolution is obtained by starting with acurve (r(φ), z(φ)) in the (x, z) plane, and rotating it around the z-axis,obtaining the parametrisation x(θ, φ) = (r(φ) cos θ, r(φ) sin θ, z(φ)).Thus we adopt the notation

u1 = θ, u2 = φ.

Theorem 6.7.1. For a surface of revolution we have Γ111 = Γ1

22 = 0,while Γ1

12 =1rdrdφ

, cf. Table 6.7.1.

Proof. For the surface of revolution

x(θ, φ) = (r(φ) cos θ, r(φ) sin θ, z(φ)), (6.7.1)

the metric coefficients are given by the matrix

(gij) =

(r2(φ) 0

0(drdφ

)2+(dzdφ

)2).

We will use the formula Γkij = 12(giℓ;j − gij;ℓ + gjℓ;i)g

ℓk (see Theo-rem 6.6.1). Since the off-diagonal coefficient g12 = 0 vanishes, thediagonal coefficients of the inverse matrix satisfy

gii =1

gii. (6.7.2)

6.8. SPHERICAL COORDINATES 67

We have∂

∂θ(gii) = 0 since the coefficients gii depend only on the vari-

able φ. Thus the terms

gii;1 = 0 (6.7.3)

vanish. Let us now compute the symbols Γ1ij for k = 1. Using formulas

(6.7.2) and (6.7.3), we obtain

Γ111 =

1

2g11(g11;1 − g11;1 + g11;1) by formula (6.7.2)

= 0 by formula (6.7.3).

Similarly,

Γ112 =

1

2g11(g11;2 − g12;1 + g12;1) =

g11;22g11

=

ddφ(r2)

2r2=r dr

dφ

r2.

Finally, Γ122 =

12g11

(g12;2 − g22;1 + g12;2) =g12;2g11

=ddφ

(0)

g11= 0. �

6.8. Spherical coordinates

Spherical coordinates are useful in understanding surfaces of revo-lution; see Section 5.10.

Definition 6.8.1. Spherical coordinates (ρ, θ, ϕ) in R3 are definedby the following formulas. We have

ρ =√x2 + y2 + z2 =

√r2 + z2

is the distance to the origin, r =√x2 + y2, while ϕ is the angle with

the z-axis, so that cosϕ = zρ. Here θ is the angle inherited from polar

coordinates in the x, y plane, so that tan θ = yxwhile x = r cos θ

and y = r sin θ.

Remark 6.8.2. The interval of definition for the variable ϕ is ϕ ∈[0, π] since ϕ = arccos z

ρand the range of the arccos function is [0, π].

Meanwhile θ ∈ [0, 2π] as usual.

Example 6.8.3. The unit sphere S2 ⊆ R3 is defined in sphericalcoordinates by the condition

S2 = {(ρ, θ, ϕ) : ρ = 1}.

68 6. Γ SYMBOLS

Definition 6.8.4. A latitude5 on the unit sphere is a circle satis-fying the equation

ϕ = constant.

Example 6.8.5. The equator of the sphere, defined by the equa-tion {ϕ = π

2}, is the only latitude that is also a great circle (see Sec-

tion 6.9) of the sphere.

Each latitude of the sphere is parallel to the equator (i.e., liesin a plane parallel to the plane of the equator). A latitude can beparametrized by setting θ(t) = t and ϕ(t) = constant.

Remark 6.8.6. On the unit sphere defined by equation {ρ = 1},at each point we have

r = sinϕ, (6.8.1)

where r is the distance from the point to the z-axis.

6.9. Great circles

In Section 8.1, we will encounter the geodesic equation. This is asystem of nonlinear second order differential equations. Our goal inthis section is to provide a geometric intuition for this equation. Wewill establish a connection between the following two items:

(1) solutions of this system, and(2) great circles on the sphere.

Such a connection is established via the intermediary of Clairaut’s re-lation for a variable point q on a great circle:

r(t) cos γ(t) = const

where r is the distance from q to the (vertical) axis of revolution and γis the angle at q between the direction of the great circle and thelatitudinal circle; cf. Theorem 7.1.2.

Remark 6.9.1. In Section 7.4, we will verify Clairaut’s relationsynthetically for great circles, and also show that the latter satisfy afirst order differential equation. In Section 8.1 we will complete theconnection by deriving Clairaut’s relation from the geodesic equation.

Let S2 ⊆ R3 be the unit sphere defined by the equation

x2 + y2 + z2 = 1.

5kav rochav

6.10. POSITION VECTOR ORTHOGONAL TO TANGENT VECTOR OF S2 69

Definition 6.9.2. A plane P through the origin is given by anequation

ax+ by + cz = 0, (6.9.1)

where a2 + b2 + c2 > 0.

Definition 6.9.3. A great circle G of S2 is given by an intersection

G = S2 ∩ P.Example 6.9.4 (A parametrisation of the equator of S2). The

equator is parametrized by

α(t) = (cos t)e1 + (sin t)e2.

Theorem 6.9.5. Every great circle can be parametrized by

α(t) = (cos t)v + (sin t)w

where v, w ∈ S2 are orthonormal vectors in R3.

Proof. The usual computation shows that α(t) is a unit vector.�

Example 6.9.6 (an implicit (non-parametric) representation of agreat circle). Recall that we have

x = r cos θ = ρ sinϕ cos θ; y = ρ sinϕ sin θ; z = ρ cosϕ . (6.9.2)

If the circle lies in the plane ax+ by+ cz = 0 where a, b, c are fixed, thegreat circle in coordinates (θ, ϕ) is defined implicitly by the equation

a sinϕ cos θ + b sinϕ sin θ + c cos θ = 0,

as in (6.9.1).

6.10. Position vector orthogonal to tangent vector of S2

Lemma 6.10.1. Let α : R → S2 be a parametrized curve on the

sphere S2 ⊆ R3. Then the tangent vectordα

dtis perpendicular to the

position vector α(t), or in formulas:⟨α(t),

dα

dt

⟩= 0.

Proof. We have 〈α(t), α(t)〉 = 1 by definition of S2. We apply theoperator d

dtto obtain

d

dt〈α(t), α(t)〉 = 0.

Next, we apply Leibniz’s rule (6.5.1) to obtain⟨α(t),

dα

dt

⟩+⟨dαdt, α(t)

⟩= 2⟨α(t),

dα

dt

⟩=

d

dt(1) = 0,

70 6. Γ SYMBOLS

completing the proof. �

CHAPTER 7

Clairaut’s relation, geodesic equation

7.1. Clairaut’s relation

Our interest in Clairaut’s relation lies in the motivation it providesfor the general geodesic equation on a surface of revolution.

Remark 7.1.1. We will use Newton’s dot notation for the deriva-tive with respect to t as in α(t) since t is not necessarily an arclengthparameter.

Theorem 7.1.2 (Clairaut’s relation). Let α(t) be a regular para-metrisation of a great circle G on S2 ⊆ R3. Let r(t) denote the dis-tance from the point α(t) to the z-axis, and let γ(t) denote the anglebetween the tangent vector α(t) to the curve and the vector tangent tothe latitude through the point α(t). Then

r(t) cos γ(t) = const.

Here the constant has value const = rmin, where rmin is the least Eu-clidean distance from a point of G to the z-axis.

7.2. Spherical sine law

Let S2 be the unit sphere in 3-space.

Definition 7.2.1. A spherical triangle is the following collectionof data: the vertices are points of S2, the sides are arcs of great circles,while the angles are defined to be the angles between tangent vectorsto the sides. Here we assume that

(1) all sides have length < π,(2) the three vertices do not lie on a common great circle.

Theorem 7.2.2 (Spherical sine law). Let a be side opposide angle α,let b be side opposite angle β, and let c be side opposite angle γ. Then

sin a

sinα=

sin b

sin β=

sin c

sin γ

Corollary 7.2.3. If γ = π2then sin a = sin c sin γ.

71

72 7. CLAIRAUT’S RELATION, GEODESIC EQUATION

Note that for small values of the sides a, c we recapture the Eu-clidean sine law

a = c sinα

as the limiting case.Some proofs are given in the footnote.1

1This material is optional. We present three proofs.

First proof of spherical sine law. Let A,B,C be the angles. We choosea coordinate system so that the three vertices of the spherical triangle are locatedat (1, 0, 0), (cos a, sina, 0) and (cos b, sin b cosC, sin b sinC). The volume of thetetrahedron formed from these three vertices and the origin is 1

6 sina sin b sinC.Since this volume is invariant under cyclic relabeling of the sides and angles,we have sin a sin b sinC = sin b sin c sinA = sin c sina sinB and therefore sinA

sin a=

sinBsin b

= sinCsin c

proving the law. See http://math.stackexchange.com/questions/

1735860. �

Second proof. Denote by A,B,C the vertices opposite the sides a, b, c. Thepoints A,B,C can be thought of as unit vectors in R3. Note that sin a = |B × C|,etc. Meanwhile sinα = |C ′ × B′| where C′ is normalisation of the orthogonalcomponent of C when the component parallel to A is eliminated (through theGram-Schmidt process). Here the component of C parallel to A is (C · A)A ofnorm cos b. The orthogonal component is C − (C · A)A is of norm sin b = |A ×C| = |C − (C · A)A|. Thus we have C ′ = C−(C·A)A

|A×C| , B′ = B−(B·A)A|A×B| . Therefore

sinαsina

=|C−(C·A)A

|A×C|×B−(B·A)A

|A×B| ||B×C| = |(C−(C·A)A)×(B−(B·A)A)|

|A×B| |A×C| |B×C| In the denominator we get

the product of the three vector products, namely |A × B| |B × C| |C × A|, whichis symmetric in the three vectors. Meanwhile in the numerator we get the norm ofthe vector

(C − (C · A)A)× (B − (B · A)A). (7.2.1)

The triple of vectors A,B,C is transformed into the triple A, (C− (C ·A)A), (B−(B · A)A) by a volume-preserving transformation. This combined with the factthat A is a unit vector shows that the norm of the vector (7.2.1) equals the absolutevalue of the determinant of the 3 × 3 matrix [ABC] (i.e., absolute value of thevolume of the parallelopiped spanned by the three vectors), which is also symmetric

in the three points. Thus we have sinαsin a

= |det[AB C] |sin a sin b sin c

, proving the spherical sinelaw. �

Third proof. There is an alternative proof that relies on the identity (a ×b) × (a × c) = (a · (b × c))a for each triple of vectors a, b, c ∈ R3. Indeed,

sinα = |(A×B)×(A×C)||A×B| |A×C| = det(AB C)

sin b sin csince A is a unit vector, and therefore

sinαsina

= det(ABC)sin a sin b sin c

as required. �

The sources for the spherical sine law are Smart 1960, pp. 9-10; Gellert et al.1989, p. 265; Zwillinger 1995, p. 469. Here (1) Gellert, W.; Gottwald, S.; Hellwich,M.; Kastner, H.; and Kunstner, H. (Eds.). “Spherical Trigonometry.” §12 in VNRConcise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold,pp. 261-282, 1989; (2) Smart, W. M. Text-Book on Spherical Astronomy, 6th ed.Cambridge, England: Cambridge University Press, 1960; (3) Zwillinger, D. (Ed.).

http://math.stackexchange.com/questions/1735860

http://math.stackexchange.com/questions/1735860

7.3. LONGITUDES 73

7.3. Longitudes

Definition 7.3.1. A longitude2 on S2 is the arc of a great circleconnecting the North Pole e3 = (0, 0, 1) and the South Pole (−1, 0, 0).

Theorem 7.3.2. A shortest path between a pair of points on S2 isan arc of great circle passing through them.

Proof. Let p0, p1 ∈ S2. Since orthogonal transformations pre-serve lengths, we can assume p0 and p1 lie on a common longitude. Tofix ideas, we will assume that this is the longitude defined by θ = 0.Thus we can assume that relative to coordinates (θ, φ) on the sphere,the point pi has coordinates (0, φi), i = 0, 1. Recall that the metriccoefficients of the sphere are g11 = sin2 φ and g22 = 1. Consider apath α(t), t ∈ [0, 1] joining the two points, so that α(0) has coordi-nates (0, φ0) and α(1) has coordinates (0, φ1). Then

∫ 1

0

∥∥∥∥dα

dt

∥∥∥∥ =

∫ 1

0

√

sin2 φ

(dθ

dt

)2

+

(dφ

dt

)2

≥∫ 1

0

∣∣∣∣dφ

dt

∣∣∣∣

≥∫ 1

0

dφ

dt

=

∫ φ1

φ0

dφ = φ1 − φ0,

proving that the arc of the longitude containing both points is a shortestpath between them. �

Lemma 7.3.3. Assume a great circle G ⊆ S2 does not pass throughthe north pole, i.e., does not contain a longitude. Let p ∈ G be thepoint with the maximal z-coordinate among the points of G. Let α(t)be a regular parametrisation of G with α(0) = p, and denote by α(0)the tangent vector to G at p. Then the great circle G has the followingthree equivalent properties:

(1) α(0) is proportional to the vector product e3 × p;(2) α(0) is perpendicular to the longitude passing through p, and(3) α(0) is tangent at p to the latitudinal circle.

“Spherical Geometry and Trigonometry.” §6.4 in CRC Standard Mathematical Ta-bles and Formulae. Boca Raton, FL: CRC Press, pp. 468-471, 1995.

2kav orech


Proof. We will prove (1). Note that 〈α(0), p〉 = 0 by Lemma 6.10.1.It remains to prove that α(0) is orthogonal to e3. By definition of thepoint p as the point with maximal z-coordinate, the function 〈α(t), e3〉achieves its maximum at t = 0. Hence we have

d

dt

∣∣∣t=0

〈α(t), e3〉 = 0. (7.3.1)

We apply Leibniz’s rule to (7.3.1) to obtain⟨dαdt

∣∣∣t=0, e3

⟩= −

⟨α(t),

de3dt

⟩= 0,

since e3 is constant. Thus 〈α(0), e3〉 = 0.3 �

7.4. Proof of Clairaut’s relation

Definition 7.4.1. The spherical distance d(p, q) on S2 is the dis-tance between p, q ∈ S2 measured along arcs of great circles:

d(p, q) = arccos〈p, q〉.As we showed in Theorem 7.3.2, d(p, q) is the least length of a path

joining p, q.

Lemma 7.4.2. Let p be the north pole: p = e3. Then the sphericaldistance d(e3, q) is the ϕ-coordinate of the point q.

Proof. This follows from the following two facts:

(1) the arclength along a unit circle equals the subtended angle;(2) that the plane containing e3 and q includes the longitude pass-

ing through the point q.

�

First we note that the complementary angle of γ is the angle withthe longitude, as follows.

Lemma 7.4.3. Let α(t) be a regular parametrisation of a great circle.If γ(t) is the angle between α(t) and (the vector tangent to) the latitude,then there is an angle of π

2− γ(t) between α(t) and the vector tangent

to the longitude at the point α(t).

3Equation of great circle. This material is optional. Note that a greatcircle on the sphere satisfies the equation cot(ϕ) = tan(γ) cos(θ − θ0), where γis the angle of inclination of the plane (see below). Indeed, a great circle is theintersection of the sphere with a plane through the origin. Let a unit normalto that plane be u = [− sin(γ), 0, cos(γ)], where for convenience we choose our xand y axes so that y = 0 (the plane contains the y-axis). Then the equation u ·[sinϕ cos θ, sinϕ sin θ, cosϕ] = 0 becomes cot(ϕ) = tan(γ) cos(θ). Rotating aroundthe z axis, this becomes cot(ϕ) = tan(γ) cos(θ − θ0) (this gamma has anything todo with the gamma from Clairaut’s relation).

7.5. DIFFERENTIAL EQUATION OF GREAT CIRCLE 75

Proof. By vanishing of the metric coefficient g12 (see Section 5.8)for a surface x(u1, u2) of revolution (see Section 6.7), the tangent vec-tors x1 = ∂x

∂θand x2 = ∂x

∂ϕare orthogonal. These are respectively the

tangents to the latitude and the longitude. Therefore the two anglesadd up to π

2. �

Proof of Clairaut’s relation. Let pmax ∈ S2 be the pointof α(t) with maximal z-coordinate. Let ϕ(t) be the spherical coordi-nate ϕ at the point α(t), so that r(t) = sinϕ(t). Consider the sphericaltriangle with vertices at the three points α(t), pmax, and the northpole e3. By Lemma 7.3.3, the angle of the triangle at pmax is π

2.

Let the arc c = c(t) be the arc of longitude joining the variablepoint α(t) to e3. Let the arc b be the arc of longitude joining pmax

to e3. By Lemma 7.4.2, the lengths b and c are respectively the ϕ-coordinates of the points pmax and α(t). Then by Corollary 7.2.3 (ofthe law of sines),

sin c sin(π2− γ)= sin b.

Note that r(t) = sin c(t) by (6.8.1). Since b is independent of t, weobtain the relation r(t) cos γ(t) = sin b, proving Clairaut’s relation. �

7.5. Differential equation of great circle

Recall that the sphere S2 is defined by the equation ρ = 1. Notethat the parametrisation in Clairaut’s formula need not be arclength.Assume that a great circle G ⊆ S2 does not pass through the Northpole. Then we can parametrize G by the value of the spherical coordi-nate θ.

Remark 7.5.1. By the implicit function theorem, we can thinkof G as defined by a suitable function

ϕ = ϕ(θ).

Theorem 7.5.2. The great circle G satisfies the following differen-tial equation for ϕ = ϕ(θ):

r2 +

(d ϕ

dθ

)2

=r4

const2, (7.5.1)

where r = sinϕ and const = sinϕmin from Theorem 7.1.2.

Proof. An element of length ds along the great circle G decom-poses into a longitudinal (along a longitude, north-south) displace-ment dϕ, and a latitudinal (east-west) displacement rdθ. These are


related byds sin γ = dϕ

ds cos γ = rdθ

so that ds2 = (dϕ)2 + (rdθ)2. Hence

rdθ tan γ = ds sin γ = dϕ,

or tan γ =dϕ

rdθ. Expressing cosine in terms of tangent, we obtain

cos2 γ =1

1 +(dϕrdθ

)2 . (7.5.2)

Substituting into (7.5.2) the value cos γ = constr

from Clairaut’s relation(Theorem 7.1.2), we obtain4

(const

r

)2(1 +

(dϕ

rdθ

)2)

= 1. (7.5.4)

Equivalently,

r2 +

(dϕ

dθ

)2

=r4

const2where r = sinϕ.

This equation is solved explicitly in terms of integrals in Example 8.1.5below. �

Remark 7.5.3. At a point where ϕ is not extremal as a functionof θ, the theorem on the uniqueness of solution applies and gives aunique geodesic through the point.

However, at a point of maximal ϕ, the hypothesis of the uniquenesstheorem does not apply. Namely, the square root expression on theright hand side of (7.5.3) does not satisfy the Lipschitz condition asthe expression under the square root sign vanishes. In fact, uniquenessfails at this point, as a latitude (which is not a geodesic) satisfies thedifferential equation, as well. Here we have r = const, and at anextremal value of ϕ one can no longer solve the equation by separationof variables (as this would involve division by the radical expressionwhich vanishes at the extremal value of ϕ). At this point, there is a

4Equivalently, 1 +(

dϕrdθ

)2=(

rconst

)2or(

dϕrdθ

)2= r2

const2− 1 or dϕ

rdθ=

√r2

const2 − 1 or 1r

dϕdθ

=√

r2

const2 − 1 or

1

sinϕ

dϕ

dθ=

√sin2 ϕ

const2− 1 (7.5.3)

7.6. METRICS CONFORMAL TO THE FLAT METRIC AND THEIR Γkij 77

Γ1ij j = 1 j = 2

i = 1 λ12λ

λ22λ

i = 2 λ22λ

−λ12λ

Γ2ij j = 1 j = 2

i = 1 −λ22λ

λ12λ

i = 2 λ12λ

λ22λ

Table 7.6.1. Symbols Γkij of a conformal metric λδij

degeneracy and general results about uniqueness of solution cannot beapplied.

7.6. Metrics conformal to the flat metric and their Γkij

A particularly important class of metrics are those conformal to theflat metric, in the following sense. We will use the symbol

λ = λ(u1, u2)

for the conformal factor of the metric, as below.

Definition 7.6.1. We say that a metric is conformal to the stan-dard flat metric if there is a function λ(u1, u2) > 0 such that gij = λδijfor all i, j = 1, 2.

Let λi =∂λ∂ui

.

Lemma 7.6.2. Consider a metric conformal to the standard flatmetric, so that gij = λ(u1, u2)δij. Then we have Γ1

11 = λ12λ, Γ1

22 = −λ12λ

,

and Γ112 =

λ22λ.

The values of the coefficients are listed in Table 7.6.1.

Proof. The general formula, as in Section 6.6, is

Γkij =1


ℓk.

For diagonal metrics this simplifies to

Γkij =1

2(gik;j − gij;k + gjk;i)g

k k

where we have underlined the index k to emphasize that no summa-tion is taking place even though k appears as both a subscript and asuperscript. Thus

Γkij =1

2gkk(gik;j − gij;k + gjk;i)


where underlining is no longer necessary since the index k appears onlyas a subscript in the formula. If i = j then the formula simplifies to

Γkii =1

2gkk(gik;i − gii;k + gik;i) =

1

2gkk(2gik;i − gii;k).

By hypothesis, we have g11 = g22 = λ(u1, u2) while g12 = 0 and thelemma follows by examining the cases. For example, let i = j = 1.Then

Γ111 =

g11;12λ

=λ12λ.

Let i = j = 2. Then

Γ122 =

2 · 0− g22;12λ

= −λ12λ.

Similar calculations yield the formulas for the coefficients Γ2ij. �

Example 7.6.3. An application to the hyperbolic metric appearsin Section 12.5.

Definition 7.6.4. Coordinates with respect to which the metric isexpressed by gij = λ(u1, u2)δij are called isothermal.

See Section 10.3 for additional details.

Corollary 7.6.5. A metric in isothermal coordinates (u1, u2) sat-isfies the identity Γ1

11 + Γ122 = 0 and similarly Γ2

11 + Γ222 = 0.

Proof. From Table 7.6.1 we have Γ111 + Γ1

22 =λ12λ

− λ12λ

= 0. Simi-

larly, Γ211 + Γ2

22 = −λ22λ

+ λ22λ

= 0. �

This corollary will be used in Section 10.3 to obtain an equationrelating the Laplacian to mean curvature.

7.7. Gravitation, ants, and geodesics on a surface

What is a geodesic on a surface?A geodesic on a surface can be thought of as the path of an ant

crawling along the surface of an apple, according to Gravitation, aPhysics textbook [MiTW73, p. 3].

Remark 7.7.1. Imagine that we peel off a narrow strip5 of theapple’s skin along the ant’s trajectory, and then lay it out flat on atable. What we obtain is a straight line, revealing the ant’s ability totravel along the shortest path.

5retzua

7.8. GEODESIC EQUATION 79

On the other hand, a geodesic is defined by a certain nonlinearsecond order ordinary differential equation, cf. (7.8.1). To make thegeodesic equation more concrete, we will examine the case of the sur-faces of revolution. Here the geodesic equation transforms into a con-servation law (conservation of angular momentum)6 called Clairaut’srelation. The latter lends itself to a synthetic verification for sphericalgreat circles, as in Theorem 7.1.2.

We will derive the geodesic equation using the calculus of varia-tions. I once heard R. Bott point out a surprising aspect of M. Morse’sfoundational work in this area. Namely, Morse systematically used thelength functional on the space of curves. The simple idea of using theenergy functional instead of the length functional was not exploiteduntil later. The use of energy simplifies calculations considerably, aswe will see in Section 9.9.2.

7.8. Geodesic equation

Consider a plane curve Rs→α

R2

(u1,u2)where α = (α1(s), α2(s)). If a

map x : R2 → R3 is a parametrisation of a surface M , the composition

Rs→α

R2

(u1,u2)→xR3

yields a curveβ = x ◦ α

on M .

Proposition 7.8.1. Every regular curve β(s) on M satisfies theidentity

β ′′ =(αi

′αj

′Γkij + αk

′′)xk +

(Lijα

j ′αi′)n

Proof. Write β = x ◦ α, then β ′ = xi(α(s))αi′ by chain rule.

Differentiatinga again, we obtain

β ′′ =d

ds(xi ◦ α)αi′ + xiα

i′′ = xijαj ′αi

′+ xkα

k′′.

Meanwhile xij = Γkijxk + Lijn. Thus

β ′′ =(Γkijxk + Lijn

)αj

′αi

′+ xkα

k′′.

Rearranging the terms proves the proposition. �

Definition 7.8.2. A curve β = x◦α is a geodesic on the surface xif the one of the following two equivalent conditions is satisfied:

6tena’ zaviti


(a) we have for each k = 1, 2,

(αk)′′

+ Γkij(αi)

′

(αj)′

= 0 where′

=d

ds, (7.8.1)

meaning that

(∀k) d2αk

ds2+ Γkij

dαi

ds

dαj

ds= 0; (7.8.2)

(b) the vector β ′′ is perpendicular to the surface and one has

β ′′ = Lijαi′αj

′n. (7.8.3)

Remark 7.8.3. The equations (7.8.1) will be derived using thecalculus of variations, in Section 9.9.2. Furthermore, by Lemma 8.1.1,such a curve β must have constant speed.

7.9. Equivalence of definitions

In Section 7.8 we defined a geodesic curve β(s) = x ◦α(s) on a sur-face with parametrisation x = x(u1, u2) in two ways that were claimedto be equivalent:

(a) we have for each k = 1, 2, (αk)′′

+ Γkij(αi)

′

(αj)′

= 0;

(b) the vector β ′′ is normal to the surface and β ′′ = Lijαi′αj

′n.

Proof of equivalence. If β satisfies (a), then applying Propo-

sition 7.8.1 to the effect that β ′′ =(αi

′αj

′Γkij + αk

′′)xk +

(Lijα

j ′αi′)n

we see that the tangent component vanishes and we obtain


′n,

showing that the vector β ′′ is normal to the surface. On the otherhand, if β ′′ is proportional to the normal vector n, then the tangentialcomponent of β ′′ must vanish, proving the equation (7.8.3) to the effectthat β ′′ = Lijα

i′αj′n. �

Corollary 7.9.1. Every geodesic in the plane is a straight line.

Proof. In the plane, all coefficients Γkij = 0 vanish. Then the

equation becomes (αk)′′

= 0. Integrating twice, we obtain that α(s) isa linear function of s. Therefore the graph is a straight line, provingthe corollary. �

CHAPTER 8

Geodesics on surface of revolution, Weingartenmap

8.1. Geodesics on a surface of revolution

Lemma 8.1.1. A curve β satisfying the geodesic equation (7.8.1) isnecessarily constant speed.

Proof. From formula (7.8.3), we have

d

ds

(‖β ′‖2

)= 2〈β ′′, β ′〉 = Lijα

i′αj′αk

′〈n, xk〉 = 0,


We use the notation u1 = θ, u2 = φ for the parameters in the caseof a surface of revolution generated by a plane curve (r(φ), z(φ)), andwrite x(θ, φ) in place of x(u1, u2). Here the function r(φ) is the distancefrom the point on the surface to the z-axis.

Lemma 8.1.2. Consider a regular curve β(s) (not necessarily geo-desic) on a surface of revolution, where s as the arclength parameter.Then the angle γ between the curve and the latitude satisfies the rela-tion cos γ = r dθ

ds.

Proof. The tangent vector to the latitude is x1 =∂x∂θ(θ, φ). Recall

that we have g11 = r2 by Theorem 6.7.1, in other words |x1| = r. Wehave

cos γ =

⟨x1|x1|

,dβ

ds

⟩.

Recall that by chain rule β ′ = x1θ′

+ x2φ′

. Thus

cos γ =1

|x1|〈x1, x1θ

′

+ x2φ′

︸︷︷︸chain rule

〉 = θ′

|x1|〈x1, x1〉︸︷︷︸

|x1|2

+φ

′

|x2|〈x1, x2〉︸︷︷︸g12=0

= θ′ |x1| = rθ

′

,


81

82 8. GEODESICS ON SURFACE OF REVOLUTION, WEINGARTEN MAP

Theorem 8.1.3. Let x = r(φ) cos θ, y = r(φ) sin θ, and z = z(φ) bethe components of a surface of revolution. Then the geodesic equationfor k = 1 is equivalent to the equation r2θ

′

= const.

Proof. The symbols for k = 1 are given by Lemma 6.7.1. Namely,

we have Γ111 = Γ1

22 = 0, while Γ112 =

drdφ

r. We will use the shorthand

notation θ(s), φ(s) respectively for the components α1(s), α2(s) of thecurve. Let ′ = d

ds. The differential equation of geodesic β = x ◦ α

for k = 1 becomes0 = θ′′ + 2Γ1

12θ′φ′

= θ′′ +2 drdφ

rθ′

φ′

.

Multiplying by r2 we obtain

0 = r2θ′′ + 2rdr

dφθ′

φ′

= (r2θ′

)′

by the Leibniz rule. Integrating (r2θ′

)′

= 0 we obtain

r2θ′

= const (8.1.1)

as required. �

Theorem 8.1.4. The geodesic equation is equivalent to Clairaut’srelation r cos γ = const; cf. Theorem 7.1.2.

Proof. Since the geodesic β is constant speed by Lemma 8.1.1, wecan assume the parameter s is arclength by rescaling the parameter by aconstant factor. Now, by Lemma 8.1.2 we have r cos γ = r2θ

′

= constfrom formula (8.1.1). Thus we obtain r(s) cos γ(s) = const, provingClairaut’s relation for an arbitrary surface of revolution. �

An integration of the geodesic equation on a surface of revolutioncan be found in the footnote.1

1This material is optional. We use the notation f(φ) = r(φ). In the caseof a surface of revolution, the geodesic equation can be integrated by means ofprimitives: 1 =

⟨dxds, dxds

⟩= 〈x1θ

′

+ x2φ′

, x1θ′

+ x2φ′〉 = g11(θ

′

)2 + g22(φ′

)2 =

f2(θ′

)2+

(dr

dφ

)2

+

(dg

dφ

)2

︸︷︷︸g22

(φ

′

)2. Thus, 1 = f2(dθds

)2+g22

(dφds

)2. Multiplying

by(dsdθ

)2, we obtain

(dsdθ

)2= f2 + g22

(dφds

dsdθ

)2= f2 + g22

(dφdθ

)2. Now from

Clairaut’s relation we have dsdθ

= f2

c, hence we obtain the formula f4

c2= f2 +

8.3. INTEGRATION IN CYLINDRICAL COORDINATES 83

8.2. Integration in polar coordinates

Following some preliminaries on areas, directional derivatives, andHessians, we will deal with a central object in the differential geom-etry of surfaces in Euclidean space, namely the Weingarten map, inSection 8.10. Spherical coordinates were reviewed in Section 6.8.

In this section, we review material from calculus on polar, cylindri-cal, and spherical coordinates as regards their role in integration. Thepolar coordinates (r, θ) in the plane arise naturally in complex analysis(of one complex variable).

Definition 8.2.1. Polar coordinates (koordinatot koteviot) (r, θ)satisfy r2 = x2 + y2 and x = r cos θ, y = r sin θ.

It is shown in elementary calculus that the area of a region D ⊆ R2

in the plane in polar coordinates is calculated using the following areaelement.

Definition 8.2.2. The area element of polar coordinates is

dA = r dr dθ.

Thus, an integral over D relative to polar coordinates is of the form∫

D

dA =

∫∫rdrdθ.

8.3. Integration in cylindrical coordinates

Cylindrical coordinates in Euclidean 3-space are studied in VectorCalculus.

Definition 8.3.1. Cylindrical coordinates (koordinatot gliliot)

(r, θ, z)

are a natural extension of the polar coordinates (r, θ) in the plane.

g22

(dφdθ

)2or 1

c2= 1

f2 + g22f4

(dφdθ

)2which is the equation (7.5.1) for a geodesic on

the sphere. Furthermore, we have dφdθ

=

√f4

c2−f2

g22= f

c

√f2−c2

( drdφ )

2+( dg

dφ)2 . Thus θ =

c∫

1f

√( dr

dφ)2+( dg

dφ )2

f2−c2dφ+ const.

Example 8.1.5. On S2, we have f(φ) = sinφ, g(φ) = cosφ, thus we

obtain θ = c∫

dφ

sinφ√

sin2 φ−c2+ const, which is the equation of a great circle

in spherical coordinates. Integrating this by a substitution u = cotφ we ob-

tain cotφ = a cos(θ − θ0) where a = 1−c2

c. See http://www.damtp.cam.ac.uk/

user/reh10/lectures/nst-mmii-handout2.pdf where θ and φ are switched.

http://www.damtp.cam.ac.uk/user/reh10/lectures/nst-mmii-handout2.pdf

http://www.damtp.cam.ac.uk/user/reh10/lectures/nst-mmii-handout2.pdf


The volume of an open region D ⊆ R3 is calculated with respect tocylindrical coordinates using a suitable volume element.

Definition 8.3.2. The volume element in cylindrical coordinatesis

dV = r dr dθ dz.

Namely, an integral is of the form∫

D

dV =

∫∫∫rdr dθ dz.

Example 8.3.3. Find the volume of a right circular cone withheight h and base a circle of radius b.

8.4. Integration in spherical coordinates

Spherical coordinates2 (ρ, θ, ϕ) in Euclidean 3-space are studied inVector Calculus.

Definition 8.4.1. Spherical coordinates (ρ, θ, ϕ) were already de-fined in Section 6.8. The coordinate ρ is the distance from the pointto the origin, satisfying

ρ2 = x2 + y2 + z2,

or ρ2 = r2 + z2, where r2 = x2 + y2. If we project the point orthog-onally to the (x, y)-plane, the polar coordinates of its image, (r, θ),satisfy x = r cos θ and y = r sin θ. The last coordinate ϕ of the spheri-cal coordinates is the angle between the position vector of the point andthe third basis vector e3 in 3-space (pointing upward along the z-axis).Thus z = ρ cosϕ while r = ρ sinϕ .

Remark 8.4.2. Here we have the bounds 0 ≤ ρ, 0 ≤ θ ≤ 2π, and0 ≤ ϕ ≤ π (note the different upper bounds for θ and ϕ).

We note the following.

(1) The area of a spherical region D ⊆ S2 on the unit sphere iscalculated using an area element sinϕ dθ d ϕ;

(2) the volume of a region in 3-space is calculated using the volumeelement of the form dV = ρ2 sinϕ dρ dθ d ϕ;

(3) the volume of a regionD ⊆ R3 is∫DdV =

∫∫∫Dρ2 sinϕ dρ dθ d ϕ.

Example 8.4.3. Calculate the volume of the spherical shell betweenspheres of radius ρ0 > 0 and ρ1 ≥ ρ0.

2koordinatot kaduriot

8.5. MEASURING AREA ON SURFACES 85

The area of a spherical region on a sphere of radius ρ = ρ1 iscalculated using the area element

dA = ρ21 sinϕ dθ d ϕ .

Thus the area of a spherical region D on a sphere of radius ρ1 is givenby the integral

∫

D

dA =

∫∫ρ21 sinϕ dθ d ϕ = ρ21

∫∫sinϕ dθ d ϕ

Example 8.4.4. Calculate the area of the spherical region on asphere of radius ρ1 contained in the first octant (so that all three Carte-sian coordinates are positive).

8.5. Measuring area on surfaces

The area of the parallelogram spanned by the tangent vectors x1and x2 can be calculated as the square root of their Gram matrix,namely the matrix of the first fundamental form. This motivates thefollowing definition.

Definition 8.5.1 (Computation of area). The area of the surfaceDparametrized by x : U → R3 is computed by integrating the area ele-ment √

g11g22 − g212 du1du2 (8.5.1)

over the domain U of the map x. Thus area(D) =∫U

√det(gij) du

1du2

where D is the region parametrized by x(u1, u2).

The presence of the square root in the formula is explained in in-finitesimal calculus in terms of the Gram matrix, cf. formula (5.7.1).

Example 8.5.2. Consider the parametrisation of the unit sphereprovided by spherical coordinates. Then the integrand is

sinϕ dθ dϕ,

and we recover the formula familiar from calculus for the area of aregion D on the unit sphere:

area(D) =

∫∫

D

sinϕ dθ dϕ .


8.6. Directional derivative

We will represent a vector v ∈ Rn as the velocity vector v = dαdt

of acurve α(t), at t = 0. Typically we will be interested in the case n = 3(or 2).

Definition 8.6.1. Given a function f of n variables, its directionalderivative3 ∇vf at a point p ∈ Rn, in the direction of a vector v isdefined by setting

∇vf =d (f ◦ α(t))

dt

∣∣∣∣t=0

where α(0) = p and α(0) = v.

Lemma 8.6.2. The definition of directional derivative is indepen-dent of the choice of the curve α(t) representing the vector v.

Proof. The lemma is proved in Elementary Calculus [Ke74]. �

Let p = x(u1, u2) be a point of a surface in R3. The tangent planeto the surface x = x(u1, u2) at the point p, denoted Tp, was definedin Section 5.7 and is the plane passing through p and spanned byvectors x1 and x2, or alternatively as the plane perpendicular to thenormal vector n at p.

Definition 8.6.3 (Orthogonal decomposition). We have an orthog-onal decomposition

R3 = Tp + Rn,

meaning that vectors v ∈ Tp and w ∈ Rn are orthogonal.

Example 8.6.4. Suppose x(u1, u2) is a parametrisation of the unitsphere S2 ⊆ R3. At a point (a, b, c) ∈ S2, the normal vector is the

position vector itself: n =

abc

. In other words, n(u1, u2) = x(u1, u2)

and we have an orthogonal decomposition R3 = Tp + Rx.

Remark 8.6.5. The sphere is defined implicitly by F (x, y, z) = 0where F (x, y, z) = x2 + y2 + z2 − 1. We have ∇F = (2x, 2y, 2z) andtherefore the gradient∇F at a point is proportional to the radius vectorof the point on the sphere (cf. Lemma 8.7.3).

3nigzeret kivunit

8.7. EXTENDING V.F. ALONG SURFACE TO AN OPEN SET IN R3 87

8.7. Extending v.f. along surface to an open set in R3

Remark 8.7.1. The notion of vector field is defined in infi 4 butonly the maslul of applied mathematics requires it. The maslul of puremathematics takes a course on rings instead.

Now let us return to the set-up

Rt→α

R2

(u1,u2)→xR3.

We consider the curve β = x ◦ α. Let v ∈ Tp be a tangent vector at a

point p ∈ M , defined by v = dβdt

∣∣t=0

, where β(0) = p. By chain rule,we have

v =dαi

dtxi.

Now consider the normal vector to the surface M

n ◦ α(t)along the curve β(t) onM . This vector is only defined along the curve.It is not defined in any open neighborhood in R3. We would like toextend it to a vector field in an open neighborhood of p ∈ R3, so as tobe able to differentiate it.

Remark 8.7.2. The curve β represents the class of curves withinitial vector v.

Lemma 8.7.3. The gradient ∇F of a function F = F (x, y, z) at apoint where the gradient is nonzero, is perpendicular to the level sur-face F (x, y, z) = 0 of the function F .

This was shown in elementary calculus.

Theorem 8.7.4. Given a regular parametrisation x(u1, u2) of Mand its normal vector n = n(u1, u2), one can extend n to a differentiablevector field N(x, y, z) defined in an open neighborhood of p ∈ M ⊆ R3,in the sense that we have

n(u1, u2) = N(x(u1, u2)

). (8.7.1)

Proof. We apply a version of the implicit function theorem forsurfaces to represent the surface implicitly by an equation F (x, y, z) =0, where

(1) the function F is defined in an open neighborhood of p ∈ M ,and

(2) ∇F 6= 0 at p.


By Lemma 8.7.3, the normalisation

N =1

|∇F |∇F

of the gradient ∇F of F gives the required extension in the senseof (8.7.1) of the normal n. �

8.8. Differentiating normal vectors n and N

We will need the directional derivative (defined in Section 8.6) inorder to define the Weingarten map in Section 8.10.

Proposition 8.8.1. Let p ∈M , and v ∈ TpM where v = β ′(0) andβ(t) = x(α(t)). Let N = N(x, y, z) be a vector field in R3 extendingthe normal vector field along the surface, namely n(u1, u2). Then thedirectional derivative ∇vN satisfies

∇vN =d (n ◦ α(t))

dt

∣∣∣∣t=0

.

Proof. By (8.7.1), the function n(t) satisfies the relation

n(α(t)) = N(β(t)),

where β = x ◦ α. The directional derivative ∇vN can be calculatedusing this particular curve β since by Lemma 8.6.2, the gradient isindependent of the choice of the curve. We therefore obtain

∇vN =d (N ◦ β(t))

dt=d (n ◦ α(t))

dt,

proving the proposition. �

8.9. Hessian of a function at a critical point

This section is intended to motivate the definition of the Weingartenmap in Section 8.10. The key observation is Remark 8.9.4 below, re-lating the Hessian and the Weingarten map.

Consider the graph in R3 of a function f(x, y) of two variables inthe neighborhood of a critical point p, where ∇f = 0.

Theorem 8.9.1. The tangent plane of the graph of f at a criticalpoint p of f is a horizontal plane.

Proof. We showed in Theorem 6.2.3 that the normal vector isn = (fx, fy,−1)t up to sign. At the critical point fx = fy = 0 andtherefore n = (0, 0, 1)t up to sign. �

8.10. DEFINITION OF THE WEINGARTEN MAP 89

The Hessian (matrix of second derivatives) of the function at thecritical point captures the main features of the local behavior of thefunction in a neighborhood of the critical point up to negligible higherorder terms. Thus, we have the following typical result concerning thesurface given by the graph of the function in R3.

Theorem 8.9.2. At a critical point p of f , assume that the eigen-values of the Hessian Hf(p) are nonzero.

(1) If the eigenvalues of the Hessian have opposite sign, then thegraph of f is a saddle point.

(2) If the eigenvalues have the same sign, the graph is a local min-imum or maximum.

Corollary 8.9.3. The determinant of the Hessian determines thenature of the critical point p. If detHf < 0 then p is a saddle point.If detHf > 0 then p is a local minimum or maximum.

Remark 8.9.4. If one thinks of the Hessian matrix Hf as defining alinear transformation (an endomorphism) of the horizontal plane Tp =R2 given by the matrix of second derivatives:

Hf : Tp → Tp

then the Hessian of a function at a critical point becomes a special caseof the Weingarten map, defined in Section 8.10.

Remark 8.9.5. The determinant of the Weingarten map plays aspecial role in determining the geometry of the surface near p, similarto that of the Hessian noted in Corollary 8.9.3.

8.10. Definition of the Weingarten map

The definition of the Weingarten operator can be viewed as ananalogue of a formula we saw in the context of curves.

Remark 8.10.1. For a plane curve α(s) with tangent vector v(s)and normal vector n(s), we had the equation

d

dsn(s) = ±kα(s)v(s); (8.10.1)

see Proposition 4.2.6. Thus, curvature is closely related to the rate ofchange of the normal vector.

The Weingarten map Wp at a point p ∈ M can be thought of as asurface analog of the formula (8.10.1) for curves. The point is that Wp

carries the information about the curvature of M .


In Section 8.9, we considered the special case of a surface given bythe graph of a function f of two variables near a critical point of f .Now consider the more general framework of a parametrized regularsurface M in R3 with a regular parametrisation x(u1, u2).

Remark 8.10.2. Instead of working with a matrix of second deriva-tives, we will give a definition of an endomorphism of the tangent plane.This is basically a coordinate-free way of talking about the Hessian ma-trix.

Using Theorem 8.7.4, we extend the normal vector field n alongthe surface to a vector field N(x, y, z) defined in an open neighborhoodof p ∈ R3, so that we have n(u1, u2) = N (x(u1, u2)).

Definition 8.10.3. Let p ∈ M . Denote by Tp = TpM the tangentplane to the surface at p. The Weingarten map (also known as theshape operator)

Wp : Tp → Tpis the endomorphism of the tangent plane given by directional deriva-tive of the vector-valued function N (extension of n as above):

Wp(v) = ∇vN =d

dt

∣∣∣∣t = 0

n ◦ α(t), (8.10.2)

where the curve β = x ◦ α is chosen so that β(0) = p while β ′(0) = v.

8.11. Properties of Weingarten map

The Weingarten map Wp : Tp → Tp at a point p ∈ M was definedin Section 8.10.

Lemma 8.11.1. The map Wp is well defined in that the image isincluded in the tangent plane Tp at p.

Proof. We have to show that the right hand side of formula (8.10.2),which is a priori a vector in R3, indeed produces a vector in the tangentplane. By Leibniz’s rule,

〈Wp(v), n(α(t))〉 = 〈∇vN, n(α(t))〉 =1

2

d

dt〈n ◦ α(t), n ◦ α(t)〉 = 0

since n is a unit vector at every point of M . Therefore the imagevector Wp(v) is orthogonal to n. Hence it lies in Tp, as required. �

CHAPTER 9

Weingarten map, Gaussian curv., second fund.form

9.1. Weingarten map is self-adjoint

Let p ∈M . Recall that the Weingarten mapWp : Tp → Tp is definedas follows. Let v ∈ Tp. Then Wp(v) = ∇vN where N = N(x, y, z) isthe extension to an open neighborhood of p ∈ R3 of the unit normalvector n = n(u1, u2) of M .

The connection with the matrix of second derivatives is given in thefollowing theorem.

Theorem 9.1.1. The operator Wp is a selfadjoint endomorphism

of the tangent plane Tp, and satisfies 〈Wp(xi), xj〉 = −⟨n, ∂2x

∂ui∂uj

⟩.

Proof. By definition of the Weingarten map, ∇vN is the deriva-tive of N along a curve with initial vector v ∈ Tp. To simplify notation,assume that p = x(0, 0). We can choose the particular “coordinate”curve of the first coordinate, γ(t) = x(t, 0). Then we have γ′(t) = x1by definition of partial derivatives. Therefore

∇x1N =d

dtN(γ(t)) =

∂n

∂u1

by definition of the directional derivative. Similarly we can show∇x2N =∂n∂u2

. Thus the coordinate lines u1 7→ x(u1, 0) and u2 7→ x(0, u2) of thechart (u1, u2) allow us to write

∇xjN =∂

∂ujn for each j = 1, 2.

Therefore we have by Leibniz rule

〈W (xi), xj〉 =⟨∂n

∂ui, xj

⟩

=∂

∂ui〈n, xj〉 − 〈n, ∂

∂uixj〉

= −⟨n,

∂2x

∂ui∂uj

⟩.

91

92 9. WEINGARTEN MAP, GAUSSIAN CURV., SECOND FUND. FORM

Thus we obtain

〈W (xi), xj〉 = −⟨n,

∂2x

∂ui∂uj

⟩. (9.1.1)

The right-hand side of (9.1.1) is an expression symmetric in i and j.Thus,

(∀i, j) 〈W (xi), xj〉 = 〈xi,W (xj)〉.This proves the selfadjointness of W by verifying it for a set of basisvectors. �

Theorem 9.1.2. The eigenvalues of the Weingarten map are real.

By Corollary 2.3.2, this is immediate from the fact that the endo-morphism Wp of the vector space TpM is selfadjoint.

9.2. Relation to the Hessian of a function

Theorem 9.2.1 (Relation to the Hessian). Given a function f oftwo variables, consider its graph x(u1, u2) = (u1, u2, f(u1, u2)). Let pbe a critical point of f . Then the inner products 〈Wp(xi), xj〉 at p formthe Hessian matrix of second derivatives Hf(p) of f at p (up to sign).

Proof. The second partial derivatives of the parametrisation arexij = (0, 0, fij)

t. The normal vector at a critical point is n = e3.Therefore we obtain 〈xij , n〉 = fij up to sign. The theorem followsfrom (9.1.1). �

Definition 9.2.2. The principal curvatures at p ∈M , denoted k1and k2, are the eigenvalues of the Weingarten map Wp.

These will be discussed in more detail in Definition 9.9.9 and Sec-tion 10.1.

Example 9.2.3. Consider the plane parametrized by x(u1, u2) =(u1, u2, 0). We have x1 = e1, x2 = e2, while the normal vector field n

n(u1, u2) =x1 × x2(x1 × x2)

= e1 × e2 = e3

is constant. It can therefore be extended to a constant vector field Ndefined in an open neighborhood in R3. Thus ∇vN ≡ 0 andWp(v) ≡ 0,and the Weingarten map is identically zero. This can be seen also fromthe vanishing of the second derivatives of the parametrisation.

In the next section we will present nonzero examples of the Wein-garten map.

9.4. WEINGARTEN MAP OF CYLINDER 93

9.3. Weingarten map of sphere

Theorem 9.3.1. Let M ⊆ R3 be the sphere of radius r > 0. Thenthe Weingarten map at every point p ∈ M of the sphere is the scalarmap Wp : Tp → Tp given by 1

rId where Id is the identity map of Tp.

Proof. Represent a tangent vector v ∈ Tp by v = β ′(0) whereβ(t) = x ◦ α(t) as usual. On the sphere of radius r > 0, we have n ◦α(t) =

1

rβ(t) (normal vector is the normalized position vector). Hence

Wp(v) = ∇vN(β(t))

=d

dt

∣∣∣∣∣t=0

n ◦ α(t)

=1

r

d

dt

∣∣∣∣∣t=0

β(t)

=1

rv.

Thus Wp(v) =1

rv for all v. In other words, Wp =

1

rId. �

Note that the Weingarten map has rank 2 in this case, i.e., it isinvertible.

9.4. Weingarten map of cylinder

Example 9.4.1 (Cylinder). For the cylinder, we have the parametri-sation

x(u1, u2) = (cosu1, sin u1, u2),

with x1 = (− sin u1, cosu1, 0)t, x2 = e3, and n = (cosu1, sin u1, 0)t. Asbefore, this can be extended to a vector field N defined in an openneighborhood in R3, with the usual relation ∇xiN = ∂

∂uin. Hence we

have

∇x1N =∂

∂u1n = (− sin u1, cosu1, 0)t. (9.4.1)

Similarly,

∇x2N =∂

∂u2n = (0, 0, 0)t. (9.4.2)

Now let v = vixi be an arbitrary tangent vector. Then

∇vN = ∇vixiN = v1∇x1N + v2∇x2N


by linearity of directional derivatives. Hence (9.4.1) and (9.4.2) yield

∇vN = v1∇x1N = v1

− sin u1

cosu1

0

,

and therefore,

Wp(v) = v1

− sin u1

cos u1

0

= v1x1,

where v = vixi. Thus, Wp(x1 + cx2) = x1 for all c ∈ R. Note that theWeingarten map has rank 1 in this case.

9.5. Coefficients Lij of Weingarten map

We have assumed throughout that the parametrisation x of M isregular. Hence the two vectors (x1, x2) from a basis of the tangentplane Tp. We can therefore introduce the following definition.

Definition 9.5.1. The coefficients Lij of the Weingarten map aredefined by setting W (xj) = Lijxi = L1

j x1 + L2j x2.

Example 9.5.2. For the plane, we have (Lij) =

(0 00 0

).

Example 9.5.3. For the sphere, we have (Lij) =

(1r

00 1

r

)= 1

r(δij).

Example 9.5.4. For the cylinder, we have L11 = 1. The remaining

coefficients vanish, so that (Lij) =

(1 00 0

).

9.6. Gaussian curvature

We continue with the terminology and notation of the previoussection. The skew-symmetrisation notation was defined in Section 1.6.

Definition 9.6.1. The Gaussian curvature function on M , de-noted K = K(u1, u2), for a surface M with a regular parametrisa-tion x(u1, u2) is the determinant of the Weingarten map:

K = det(Wp).

Remark 9.6.2. In practice we calculate K using the formula K =det(Lij) = L1

1L22 − L1

2L21 = 2L1

[1L22].

9.7. SECOND FUNDAMENTAL FORM 95

Remark 9.6.3. The signed curvatures of oriented curves will bedefined in Section 11.2 and discussed in Section 11.7. Then we will beable to assert that Gaussian curvature K of a surface is the product

K = k1 · k2of signed curvatures k1 and k2 of the pair of curves M ∩P1 andM ∩P2,where the planes Pi are defined by Pi = Span(n, vi) and v1, v2 areorthogonal eigenvectors of the Weingarten map. We will mostly workwith Definition 9.6.1 of Gaussian curvature; we include the additionaldefinition in terms of signed curvatures of curves for general culture asit was the original definition by Gauss.

Example 9.6.4. Cylinder, plane K = 0, cf. Figure 12.2.1. For a

sphere of radius r, we have K = det

(1r

00 1

r

)=

1

r2.

Remark 9.6.5 (Sign of Gaussian curvature). Of particular geomet-ric significance is the sign of the Gaussian curvature. The geometricmeaning of negative Gaussian curvature is a saddle point. The geo-metric meaning of positive Gaussian curvature is a point of convexity,such as local minimum or local maximum of the graph of a function oftwo variables.

9.7. Second fundamental form

Recall that we extended the normal vector field n(u1, u2) along thesurface M near a point p ∈ M , to a smooth vector field N(x, y, z)defined in an open neighborhood of p ∈ R3.

Definition 9.7.1. The second fundamental form IIp is the bilinearform on the tangent plane Tp defined for u, v ∈ Tp by

IIp(u, v) = −〈∇uN, v〉. (9.7.1)

Remark 9.7.2. The second fundamental form measures the curva-ture of geodesics on x viewed as curves of R3 (cf. 4.1.3) as illustratedby Theorem 9.8.1 below.

Remark 9.7.3. The sign in formula (9.7.1) is just a convention, tomake some later formulas, such as (9.7.2), to work out better. Thissign does not affect the sign of the Gaussian curvature.

Definition 9.7.4. The coefficients Lij of the second fundamentalform are defined to be

Lij = IIp(xi, xj) = −⟨∂n

∂ui, xj

⟩.


Lemma 9.7.5. The coefficients Lij of the second fundamental formare symmetric in i and j, more precisely

Lij = +〈xij, n〉. (9.7.2)

Proof. We have 〈n, xi〉 = 0. Hence ∂∂uj

〈n, xi〉 = 0, i.e.⟨

∂

∂ujn, xi

⟩+ 〈n, xij〉 = 0

or〈n, xij〉 = −〈∇xjN, xi〉 = +II(xj , xi)

and the proof is concluded by the equality of mixed partials. �

Lemma 9.7.6. The second fundamental form allows us to identifythe normal component of the second partial derivatives of the parametri-sation x; namely we have

xij = Γkijxk + Lijn.

Proof. We write xij = Γkijxk + cn. Now form the inner productwith n to obtain Lij = 〈xij , n〉 = 0 + c〈n, n〉 = c. �

9.8. Geodesics and second fundamental form

Theorem 9.8.1. Let β(s) be a unit speed geodesic on a surfaceM ⊆R3, so that β ′(s) ∈ TpM at a point p = β(s). Then the absolute valueof the second fundamental form applied to β ′ is the curvature of β at p:

|IIp(β ′, β ′)| = kβ(s),

where kβ is the curvature of β viewed as a curve in R3.

Proof. We apply formula (7.8.3). Since |n| = 1, we have

kβdef= |β ′′| =

∣∣∣Lijαi′αj ′∣∣∣ .

On the other hand. recall that β = x ◦ α. We have

II(β ′, β ′) = II(xiαi′, xjα

j ′) = αi′αj

′II(xi, xj) = αi

′αj

′Lij ,

completing the proof. �

Proposition 9.8.2. We have the following relation between thecoefficients of the Weingarten map and the second fundamental form:

Lij = −Lkjgki.Proof. By definition,

Lij = 〈xij , n〉 = −⟨

∂

∂ujn, xi

⟩= −〈Lkjxk, xi〉 = −Lkjgki


9.9. THREE FORMULAS FOR GAUSSIAN CURVATURE 97

Corollary 9.8.3. We have the following relation between the co-efficients of the Weingarten map and the coefficeints of the second fun-damental form: Li j = −gikLkj.

Proof. We start with the relation Lij = −Lkjgki of Proposi-tion 9.8.2, multiply it on both sides by giℓ, and sum over i, obtaining

Lijgiℓ = −Lkjgkigiℓ

= −Lkjδℓk= −Lℓj

as required. �

9.9. Three formulas for Gaussian curvature

Theorem 9.9.1. We have the following three equivalent formulasfor the Gaussian curvature:

(a) K = det(Lij) = 2L1[1L

22];

(b) K = det(Lij)/det(gij);(c) K = − 2

g11L1[1L

22].

Proof. Here formula (a) is our definition of K.We will continue making use of the Einstein summation convention,

i.e. suppress the summation symbol∑

when the summation indexoccurs simultaneously as a subscript and a superscript in a formula,cf. (10.7.1). To prove formula (b), we use the formula Lij = −Lkjgki ofProposition 9.8.2. By the multiplicativity of determinant with respectto matrix multiplication,

det(Lij) = (−1)2det(Lij)

det(gij).

Let us now prove formula (c). The proof is a calculation. Note that bydefinition, 2L1[1L

22] = L11L

22 − L12L

21. Hence

− 2

g11

(L1[1L

22]

)=

1

g11

((L1

1g11 + L21g21

)L2

2 −(L1

2g11 + L22g21

)L2

1

)

=1

g11

(g11L

11L

22 + g21L

21L

22 − g11L

12L

21 − g21L

22L

21

)

= det(Li j) = K

as required. �

Remark 9.9.2. We can either calculate using the formula Lij =−Lkjgki, or the formula Lij = −Lkigkj. Only the former one leads tothe appropriate cancellations as above.


9.9.1. Normal and geodesic curvatures. The material in this sec-tion is optional. Let x(u1, u2) be a surface in R3. Let α(s) = (α1(s), α2(s))a curve in R2, and consider the curve β = x ◦ α on the surface x. Con-sider also the normal vector to the surface, denoted n. The tangent unit

vector β′ = dβds is perpendicular to n, so β′, n, β′ × n are three unit vector

spanning R3. As β′ is a unit vector, it is perpendicular to β′′, and there-fore β′′ is a linear combination of n and β′×n. Thus β′′ = knn+ kg(β

′×n),where kn, kg ∈ R are called the normal and the geodesic curvature (resp.).Note that kn = β′′ · n, kg = β′′ · (β′ ×n). If k is the curvature of β, then wehave that k2 = ||β′′||2 = k2n + k2g .

Exercise 9.9.3. Let β(s) be a unit speed curve on a sphere of radius r.Then the normal curvature of β is ±1/r.

Remark 9.9.4. Let π be a plane passing through the center of the sphereS in Exercise 9.9.3, and Let C = π ∩ S. Then the curvature k of C is 1/r,and thus kg = 0. On the other hand, if π does not pass through the centerof S (and π ∩ S 6= ∅) then the geodesic curvature of the intersection 6= 0.

Proposition 9.9.5. If a unit speed curve β on a surface is geodesic thenits geodesic curvature is kg = 0.

Proof. If β is a geodesic, then β′′ is parallel to the normal vector n, soit is perpendicular to n× β′′, and therefore kg = β′′ · (n× β′′) = 0. �

Exercise 9.9.6. The inverse direction of Proposition 9.9.5 is also cor-rect. Prove it.

Example 9.9.7. Any line γ(t) = at + b on a surface is a geodesic, asγ′′ = 0 and therefore kg = 0.

Example 9.9.8. Take a cylinder x of radius 1 and intersect it with aplane π parallel to the xy plane. Let C = x ∩ π - it is a circle of radius 1.Thus k = 1. Show that kn = 1 and thus C is a geodesic.

Definition 9.9.9. The principal curvatures, denoted k1 and k2, are theeigenvalues of the Weingarten map W .

Assume k1 > k2.

Theorem 9.9.10. The minimal and maximal values of the absolute valueof the normal curvature |kn| at a point p of all curves on a surface passingthrough p are |k2| and |k1|.

Proof. This is proven using the fact that kg = 0 for geodesic curvesand from Theorem 3.9.4?1 �

1See which theorem this is.

9.9. THREE FORMULAS FOR GAUSSIAN CURVATURE 99

9.9.2. Calculus of variations and the geodesic equation. Thematerial in this subsection is optional. Calculus of variations is known astachsiv variatsiot. Let α(s) = (α1(s), α2(s)), and consider the curve β =x ◦ α on the surface given by [a, b]s→αR2 →xR3. Consider the energy

functional E(β) =∫ ba ‖β′‖2ds. Here ′ = d

ds . We have dβds = xi(α

i)′

by chainrule. Thus

E(β) =∫ b

a〈β′, β′〉ds =

∫ b

a

⟨xiα

i′, xjαj ′⟩ds =

∫ b

agijα

i′αj′ds. (9.9.1)

Consider a variation α(s) → α(s)+ tδ(s), t small, such that δ(a) = δ(b) = 0.If β = x ◦α is a critical point of E , then for any perturbation δ vanishing atthe endpoints, the following derivative vanishes:

0 =d

dt

∣∣∣∣∣t = 0

E(x ◦ (α+ tδ))

=d

dt

∣∣∣∣∣t = 0

{∫ b

agij(α

i + tδi)′

(αj + tδj)′

ds

}(from equation (9.9.1))

=

∫ b

a

∂ (gij ◦ (α + tδ))

∂tαi

′

αj′

ds

︸︷︷︸A

+

∫ b

agij(αi

′

δj′

+ αj′

δi′)ds

︸︷︷︸B

,

so that we have

A+B = 0. (9.9.2)

We will need to compute both the t-derivative and the s-derivative of thefirst fundamental form. The formula is given in the lemma below.

Lemma 9.9.11. The partial derivatives of gij = gij ◦ (α(s) + tδ(s))

along β = x ◦ α are given by the following formulas: ∂∂t(gij ◦ (α + tδ)) =

(〈xik, xj〉+ 〈xi, xjk〉)δk, and ∂gik∂s = (〈xim, xk〉+ 〈xi, xkm〉)(αm)

′

.

Proof. We have

∂

∂t(gij ◦ (α+ tδ)) =

∂

∂t〈xi ◦ (α+ tδ)), xj ◦ (α+ tδ)〉

=

⟨∂

∂t(xi ◦ (α+ tδ)), xj

⟩+

⟨xi,

∂

∂t(xj ◦ (α+ tδ))

⟩

= 〈xikδk, xj〉+ 〈xi, xjkδk〉 = (〈xik, xj〉+ 〈xi, xjk〉)δk.Furthermore, g′ik = 〈xi, xk〉′ = 〈xim(αm)′ , xk〉+ 〈xi, xkm(αm)

′〉. �

Lemma 9.9.12. Let f ∈ C0[a, b]. Suppose that for all g ∈ C0[a, b] we

have∫ ba f(x)g(x)dx = 0. Then f(x) ≡ 0. This conclusion remains true if

we use only test functions g(x) such that g(a) = g(b) = 0.

Proof. We try the test function g(x) = f(x). Then∫ ba (f(x))

2ds = 0.

Since (f(x))2 ≥ 0 and f is continuous, it follows that f(x) ≡ 0. If we


want g(x) to be 0 at the endpoints, it suffices to choose g(x) = (x− a)(b −x)f(x). �

Theorem 9.9.13. Suppose β = x ◦ α is a critical point of the en-ergy functional (endpoints fixed). Then β satisfies the differential equation

(∀k) (αk)′′+ Γkij(α

i)′(αj)

′= 0.

Proof. We use Lemma 9.9.11 to evaluate the termA from equation (9.9.2)

as follows: A =∫ ba (〈xik, xj〉+〈xi, xjk〉)(αi)

′

(αj)′

δkds = 2

∫ b

a〈xik, xj〉αi

′

αj′

δkds,

since summation is over both i and j. Similarly, B = 2∫ ba gijα

i′

δj′

ds =

−2∫ ba

(gijα

i′)′

δjds by integration by parts, where the boundary term van-

ishes since δ(a) = δ(b) = 0. Hence B = −2∫ ba

(gikα

i′)′δkds by changing an

index of summation. Thus 12ddt

∣∣∣∣∣t=0

(E) =∫ ba

{〈xik, xj〉αi′αj ′ −

(gikα

i′)′}

δkds.

Since this is true for any variation (δk), by Lemma 9.9.12 we obtain the

Euler-Lagrange equation (∀k) 〈xik, xj〉αi′αj ′ −(gikα

i′)′

≡ 0, or

〈xik, xj〉αi′αj ′ − gik′αi

′ − gikαi′′ = 0. (9.9.3)

Using the formula from Lemma 9.9.11 for the s-derivative of the first fun-damental form, we can rewrite the formula (9.9.3) as follows:

0 = 〈xik, xj〉αi′

αj′

− 〈xim, xk〉αm′

αi′

− 〈xi, xkm〉αi′

αm′ − gikα

i′′

= −〈Γnimxn, xk〉αm′

αi′

− gikαi′′

= −Γnimgnkαm′

αi′

− gikαi′′

,

where the cancellation of the first and the third term in the first line resultsfrom replacing index i by j and m by i in the third term. This is true ∀k.Now multiply by gjk to obtain gjkΓnimgnkα

m′

αi′

+gjkgikαi′

= δjnΓnimαm′

αi′

+

δjiαi′′

= Γjimαm′

αi′

+ αj′′

= 0, which is the desired geodesic equation. �

CHAPTER 10

Principal curvatures, minimal surf., thm.egregium

10.1. Principal curvatures of surfaces

In Section 8.10 we defined the Weingarten map Wp : Tp → Tp whichis an endomorphism of the tangent plane Tp at a point p ∈ M of asurface M .

Definition 10.1.1. The principal curvatures at the point p, de-noted k1 and k2, are the eigenvalues of the Weingarten map Wp.

Example 10.1.2. LetM be the hyperbolic paraboloid given by thegraph in R3 of the function f(x, y) = ax2−by2. The origin p = (0, 0) isa critical point of f . Therefore the matrix of the Weingarten map Wp

at p is the Hessian matrix Hf of f , namely

(2a 00 −2b

). Therefore the

principal curvatures at the origin are k1 = a and k2 = −b (the order ofthe two eigenvalues is immaterial).

Remark 10.1.3. The curvatures k1 and k2 are necessarily real. Thisfollows from the selfadjointness of W (Theorem 9.1.1) together withCorollary 2.3.2.

Theorem 10.1.4. The Gaussian curvature K(u1, u2) = det(Wp) ata point p = x(u1, u2) equals the product of the principal curvatures at p.

Proof. The determinant of a 2 by 2 matrix equals the productof its eigenvalues: K = det(Lij) = k1k2, where (Lij) is the matrixrepresenting the endomorphism Wp. �

Recall that IIp denotes the second fundamental form at p ∈M . Weproved in Theorem 9.8.1 that

|IIp(β ′, β ′)| = kβ, (10.1.1)

where kβ is the curvature of the unit speed geodesic β(s) with velocityvector β ′.

Theorem 10.1.5. Let v ∈ TpM be a unit eigenvector belonging to aprincipal curvature k at the point p ∈M . Let β(s) be a geodesic on M

101

102 10. PRINCIPAL CURVATURES, MINIMAL SURF., THM. EGREGIUM

satisfying β ′(0) = v. Then the curvature of β as a space curve is theabsolute value of k :

kβ(0) = |k|.This will be proved following Remark 10.1.7 below.

Corollary 10.1.6. The absolute value of the Gaussian curvatureat a point p of a regular surface in R3 is the product of curvatures oftwo perpendicular geodesics passing through p, whose tangent vectorsare eigenvectors of the Weingarten map at the point.

Remark 10.1.7. The shortcoming of this corollary is the fact thatat this point we are only able to express the absolute value |K| interms of curvatures of curves through p. This will be rectified whenwe introduce a finer invariant called signed curvature of an orientedplane curve in Section 11.2. Signed curvatures will be discussed inSection 11.7.

Proof of Theorem 10.1.5. Since v is an eigenvector of Wp, weobtain from (10.1.1) that

kβ(0) = |IIp(β ′, β ′)|= |⟨Wp (β

′), β ′⟩ |= |〈kβ ′, β ′〉|= |k〈β ′, β ′〉|= |k|


10.2. Mean curvature, minimal surfaces

Definition 10.2.1. The mean curvature H of the surface M at apoint p is half the trace of the Weingarten map: H = 1

2trace(Wp) =

1

2(k1 + k2) =

1

2Lii.

Definition 10.2.2. A surface M is called minimal if H = 0 atevery point, i.e. k1 + k2 = 0.

Remark 10.2.3. The sign of the mean curvature has no geomet-ric meaning and depends on the choice of normal vector n (from thepair n,−n) used in the definition ofWp and IIp. This is in contrast withGaussian curvature ofM where the sign does have geometric meaning.

See Table 12.2.1 on plane and cylinder for comparison of the geo-metric meaning of mean curvature and Gaussian curvature.

10.2. MEAN CURVATURE, MINIMAL SURFACES 103

Figure 10.2.1. Scherk’s minimal surface

Lemma 10.2.4. Let f(x) = ln cos x and h(x) = f ′(x). Then we

have 1+h2(x)f ′′(x)

= −1 identically in x.

Proof. If f(x) = ln cosx then h(x) = f ′(x) = − tan x and f ′′(x) =−1

cos2 x. Therefore 1+h2(x)

f ′′(x)= −(1 + tan2 x) cos2 x = −1 as required. �

Theorem 10.2.5 (Scherk surface). The Scherk surface, parametrizedby(x, y, ln

(cos ycos x

)), is a minimal surface.

Proof. Let f(x) = ln cosx. Then the Scherk surface is by defi-nition x(x, y) = (x, y, f(y)− f(x)). Clearly, we have x12 = 0. There-fore L12 = 0. Thus the matrix (Lij) is diagonal. By Corollary 9.8.3,the mean curvature H satisfies 2H = traceWp = L11g

11 + L22g22 and

therefore the condition H = 0 is equivalent to L11g11 + L22g

22 = 0,where g11 = 1

g11and g22 = 1

g22. Multiplying by g11g22, we obtain an

equivalent condition:

H = 0 ⇐⇒ L11g22 + L22g11 = 0. (10.2.1)

Let h(x) = f ′(x). We have x1 = (1, 0,−h(x))t and x2 = (0, 1, h(y))t.Hence g11 = 1 + h2(x) and g22 = 1 + h2(y). The normal vector isthe normalisation of the cross product (−h(x), h(y), 1)t. Let C =√1 + h2(x) + h2(y), so that

n =1

C(h(x),−h(y), 1)t.


Figure 10.2.2. Helicoid: a minimal surface

Note that x11 = (0, 0,−f ′′(x))t. Then

L11 = 〈n, x11〉 = −f′′(x)

C

and similarly L22 =f ′′(y)C

. Thus by (10.2.1) we have

H = 0 ⇐⇒ f ′′(x)(1 + h2(y)) = f ′′(y)(1 + h2(x)),

or equivalentlyf ′′(x)

1 + h2(x)=

f ′′(y)

1 + h2(y),

and we conclude by Lemma 10.2.4. See Figure 10.2.1. �

Remark 10.2.6. Geometrically, a minimal surface is representedlocally by a soap film.1 See Figure 10.2.2 for another example (helicoid).

10.3. Minimal surfaces in isothermal coordinates

In this section we study the PDE of a minimal surface.

Definition 10.3.1. A parametrisation x(u1, u2) is called isother-mal if there is a function f = f(u1, u2) > 0 such that gij = f 2δij,i.e. 〈x1, x1〉 = 〈x2, x2〉 and 〈x1, x2〉 = 0.

1krum sabon, as opposed to bu’at sabon. Dip (tvol) a wire (tayil) into soapywater.

10.3. MINIMAL SURFACES IN ISOTHERMAL COORDINATES 105

Definition 10.3.2. The function f 2 is called conformal factor ofthe metric.

Proposition 10.3.3. Assume x(u1, u2) is a parametrisation in isother-mal coordinates. Then it satisfies the partial differential equation

∂2x

∂(u1)2+

∂2x

∂(u2)2= −2f 2Hn,

where n is the normal vector.

Proof. We have

∆ x = x11 + x22

= Γ111x1 + Γ2

11x2 + L11n+ Γ122x1 + Γ2

22x2 + L22n

=(Γ111 + Γ1

22

)x1 +

(Γ211 + Γ2

22

)x2 + (L11 + L22)n.

By Corollary 7.6.5, with respect to isothermal coordinates we nec-essarily have Γ1

11 + Γ122 = 0 and Γ2

11 + Γ222 = 0. Recall that with

respect to isothermal coordinates, we have Lii = −f 2Lii. There-

fore ∆x = (L11 +L22)n = −(L11 +L2

2)f2n = −2Hf 2n as required. �

10.3.1. Alternative proof. This subsection is optional. Note that Lij =

−Lmjgmi = −Lmjf2δmi = −f2Lij . Thus Lij = −Lijf2

so that the mean cur-

vature H satisfies

H =1

2Lii = −L11 + L22

2f2. (10.3.1)

Differentiating 〈x1, x2〉 = 0 we obtain ∂∂u2

〈x1, x2〉 = 0. Therefore

〈x12, x2〉+ 〈x1, x22〉 = 0. (10.3.2)

So −〈x12, x2〉 = 〈x1, x22〉. By Definition 10.3.1 of isothermal coordinates,we have

〈x1, x1〉 − 〈x2, x2〉 = 0. (10.3.3)

Differentiating (10.3.3) and applying (10.3.2), we obtain 0 = ∂∂u1

〈x1, x1〉 −∂∂u1

〈x2, x2〉 = 2〈x11, x1〉 − 2〈x21, x2〉 = 2〈x11, x1〉 + 2〈x22, x1〉 = 2〈x11 +

x22, x1〉. Inspecting the u2-derivatives, we similarly obtain 〈x11 + x22, x2〉 =0. Since x1, x2 and n form an orthogonal basis, the sum x11 + x22 is

proportional to n. Write x11 + x22 = cn. Applying (10.3.1), we ob-

tain c = 〈x11 + x22, n〉 = 〈x11, n〉 + 〈x22, n〉 = L11 + L22 = −2f2H, as

required.


10.4. Minimality and harmonic functions

In this section we study the relation between minimal surfaces andharmonic functions. The latter are familiar from complex functiontheory. We first recall a definition from Section 4.5.

Definition 10.4.1. Let F : R2 → R, (u1, u2) 7→ F (u1, u2). TheLaplacian, denoted ∆(F ), is the second-order differential operator onfunctions, defined by

∆F =∂2F

∂(u1)2+

∂2F

∂(u2)2.

Functions in the kernel of the Laplacian are called harmonic:

Definition 10.4.2. We say that F is harmonic if ∆F = 0.

Harmonic functions are important in the study of heat flow, or heattransfer.2

Theorem 10.4.3. Let x(u1, u2) = (x(u1, u2), y(u1, u2), z(u1, u2))in R3 be a parametrisation of a surface M . Assume that the coor-dinates (u1, u2) are isothermal. Then M is minimal if and only if thecoordinate functions x, y, z are harmonic.

Proof. From Proposition 10.3.3 we have

‖∆(x)‖ =√

(∆x)2 + (∆y)2 + (∆z)2 = 2f 2|H|.Therefore the Laplacian of x vanishes if and only if H = 0. �

Example 10.4.4. Let a > 0. The catenoid is parametrized asfollows:

x(θ, φ) = (a coshφ cos θ, a coshφ sin θ, aφ).

Here r(φ) = a coshφ, while z(φ) = aφ. Then g11 = a2 cosh2 φ = r2.Also, g22 = (a sinhφ)2 + a2 = a2 cosh2 φ = r2. Finally, g12 = 0. Weconclude that the coordinates are isothermal. Finally,

x11 + x22 = (−a cosh φ cos θ,−a cosh φ sin θ, 0)+ (a coshφ cos θ, a coshφ sin θ, 0)

= (0, 0, 0).

By Theorem 10.4.3, the catenoid is a minimal surface. In fact, it is theonly surface of revolution which is minimal [We55, p. 179].

2ma’avar chom.

10.5. INTRODUCTION TO THEOREMA EGREGIUM; INTRINSIC VS EXTRINSIC107

10.5. Introduction to theorema egregium; intrinsic vs

extrinsic

Some of the material in this chapter has already been covered inearlier chapters. We include such material here for the benefit of thereader primarily interested in understanding Gaussian curvature.

The present Chapter is an introduction to Gaussian curvature. Un-derstanding the intrinsic nature of Gaussian curvature, i.e. the theo-rema egregium of Gauss, clarifies the geometric classification of sur-faces. A historical account and an analysis of Gauss’s proof of thetheorema egregium may be found in [Do79].3

We present a compact formula for Gaussian curvature and its proof,along the lines of the argument in M. do Carmo’s book [Ca76]. Theappeal of an old-fashioned, computational, coordinate notation proof isthat it obviates the need for higher order objects such as connections,tensors, exponental map, etc, and is, hopefully, directly accessible to astudent not yet familiar with the subject, cf. Remark 12.5.5.

Remark 10.5.1. We would like to distinguish two types of proper-ties of a surface in Euclidean space: intrinsic and extrinsic.

Understanding the extrinsic/intrinsic dichotomy is equivalent to un-derstanding the theorema egregium of Gauss. The theorema egregium isthe key insight lying at the foundation of differential geometry as con-ceived by Bernhard Riemann in his essay [Ri1854] presented beforethe Royal Scientific Society of Gottingen in 1854 (see Section 10.6).

The theorema egregium asserts that an infinitesimal invariant of asurface in Euclidean space, called Gaussian curvature, is an “intrinsic”invariant of the surface M . In other words, Gaussian curvature of Mis independent of its isometric embedding in Euclidean space. Thistheorem paves the way for an intrinsic definition of curvature in modernRiemannian geometry.

Remark 10.5.2. We will prove that Gaussian curvature K is anintrinsic invariant of a surface in Euclidean space, in the followingprecise sense: K can be expressed in terms of the coefficients of thefirst fundamental form and their derivatives alone.

A priori the possibility of thus expressing K is not obvious, as thenaive definition ofK involves the coefficients of the second fundamentalform (alternatively, of the Weingarten map).

3Our formula (10.9.2) for Gaussian curvature is similar to the traditional for-mula for the Riemann curvature tensor in terms of the Levi-Civita connection (notethe antisymmetrisation in both formulas, and the corresponding two summands),without the burden of the connection formalism.


10.6. Riemann’s formula

The intrinsic nature of Gaussian curvature paves the way for a tran-sition from classical differential geometry, to a more abstract approachof modern differential geometry (see also Subsection 14.1). The dis-tinction can be described roughly as follows. Classically, one studiessurfaces in Euclidean space. Here the first fundamental form (gij) ofthe surface is the restriction of the Euclidean inner product. Mean-while, abstractly, a surface comes equipped with a set of coefficients,which we deliberately denote by the same letters, (gij) in each coordi-nate patch, or equivalently, its element of length. One then proceedsto study its geometry without any reference to a Euclidean embedding,cf. (17.12.2).

Such an approach was pioneered in higher dimensions in Riemann’sessay. The essay contains a single formula [Ri1854, p. 292]. See Spivak[Sp79, p. 149]), where the formula appears on page 159. This is theformula for the element of length of a surface of constant (Gaussian)curvature K ≡ α:

1

1 + α4

∑x2

√∑dx2 (10.6.1)

where today, of course, we would incorporate a summation index i aspart of the notation, as in

1

1 + α4

∑i(x

i)2

√∑

i

(dxi)2 (10.6.2)

cf. formulas (17.12.3), (17.25.1), and (19.3.1).We will explain the notation dxi in Section 14.6.Given a surface M in 3-space which is the graph of a function of

two variables, consider a critical point p ∈M . The Gaussian curvatureof the surface at the critical point p is the determinant of the Hessianof the function, i.e. the determinant of the two-by-two matrix of itssecond derivatives.

The implicit function theorem allows us to view any point of aregular surface, as such a critical point, after a suitable rotation. Wehave thus given the simplest possible definition of Gaussian curvatureat any point of a regular surface.

Remark 10.6.1. The appeal if this definition is that it allows oneimmediately to grasp the basic distinction between negative versuspositive curvature, in terms of the dichotomy “saddle point versuscup/cap”.

10.7. REVIEW OF NOTATION 109

It will be more convenient to use an alternative definition in termsof the Weingarten map, which is readily shown to be equivalent to thedefinition in terms of the Hessian.

10.7. Review of notation

We denote the coordinates in R2 by (u1, u2). In R3, let 〈 , 〉 be theEuclidean inner product. Let x = x(u1, u2) : R2 → R3 be a regularparametrized surface. Here “regular” means that the vector valuedfunction x has Jacobian of rank 2 at every point. Consider the followingdata.

(1) We have the partial derivatives xi =∂∂ui

(x), where i = 1, 2.Thus, vectors x1 and x2 form a basis for the tangent plane atevery point.

(2) We let xij =∂2x

∂ui∂uj∈ R3.

(3) We let n = n(u1, u2) be a unit normal to the surface at thepoint x(u1, u2), so that 〈n, xi〉 = 0.

(4) The first fundamental form (gij) is given in coordinates by gij =〈xi, xj〉.

(5) The second fundamental form (Lij) is given in coordinatesby Lij = 〈n, xij〉.

(6) The symbols Γkij are uniquely defined by the decomposition

xij = Γkijxk + Lijn

= Γ1ijx1 + Γ2

ijx2 + Lijn,(10.7.1)

where the repeated (upper and lower) index k implies summa-tion, in accordance with the Einstein summation convention.

(7) The Weingarten map (Lij) is an endomorphism of the tangentplane Rx1 +Rx2. It is uniquely defined by the decomposition

nj = Lijxi

= L1jx1 + L2

jx2,

where nj =∂∂uj

(n).

(8) We have the relation Lij = −Lki gjk.(9) We use the notation Γkij;ℓ for the ℓ-th partial derivative of the

symbol Γkij.(10) We will denote by square brackets [ ] the antisymmetrisa-

tion over the pair of indices found in between the brackets,e.g. a[ij] =

12(aij − aji).

(11) We have g[ij] = 0, L[ij] = 0, Γk[ij] = 0, and x[ij] = 0.


10.8. An identity involving the Γs and the Ls

Lemma 10.8.1. The third partials of x satisfy

xijk = (xij)k = (xik)j.

Proof. This is immediate from the equality of mixed partials. �

The following technical result will yield the theorema egregium asan easy consequence.

Proposition 10.8.2. We have the following relation

Γqi[j;k] + Γmi[j Γqk]m = −Li[jLqk]

for each set of indices i, j, k, q (with, as usual, an implied summationover the index m).

Proof. Consider the third partial derivative xijk =∂3x

∂ui∂uj∂uk. Let

us calculate its tangential component relative to the basis {x1, x2, n}.Recall that nk = Lpkxp and xjk = Γℓjkxℓ + Likn. Thus, we have

(xij)k =(Γmijxm + Lijn

)k

= Γmij;kxm + Γmijxmk + Lijnk + Lij;kn

= Γmij;kxm + Γmij

(Γpmkxp + Lmkn

)+ Lij (L

pkxp) + Lij;kn.

Grouping together the tangential terms, we obtain

(xij)k = Γmij;kxm + Γmij

(Γpmkxp

)+ Lij (L

pkxp) + (. . .)n

=(Γqij;k + ΓmijΓ

qmk + LijL

qk

)xq + (. . .)n

=(Γqij;k + ΓmijΓ

qkm + LijL

qk

)xq + (. . .)n,

since the symbols Γqkm are symmetric in the two subscripts.By Lemma 10.8.1, the symmetry in j, k (equality of mixed partials)

implies the following identity: xi[jk] = 0. Therefore

0 = xi[jk]

= (xi[j)k]

=(Γqi[j;k] + Γmi[jΓ

qk]m + Li[jL

qk]

)xq + (. . .)n,

and therefore Γqi[j;k] + Γmi[jΓqk]m + Li[jL

qk] = 0 for each q = 1, 2. �

10.9. THE THEOREMA EGREGIUM OF GAUSS 111

10.9. The theorema egregium of Gauss

Recall from Section 9.9 that we have K = − 2g11L1[1L

22], where by

definitionK = det(Lij) = 2L1

[1L22]. (10.9.1)

Theorem 10.9.1 (Theorema egregium). The Gaussian curvaturefunction K = K(u1, u2) of a surface can be expressed in terms of thecoefficients of the first fundamental form alone (and their first andsecond derivatives) as follows:

K =2

g11

(Γ21[1;2] + Γj1[1Γ

22]j

), (10.9.2)

where the symbols Γkij can be expressed in terms of the derivatives of gijbe the formula Γkij =


ℓk, where (gij) is the inversematrix of (gij).

Proof of theorema egregium. We present a streamlined version ofdo Carmo’s proof [Ca76, p. 233]. The proof is in 3 steps.

(1) We express the third partial derivative xijk in terms of both theΓ’s (intrinsic information) and the L’s (extrinsic information).

(2) The equality of mixed partials yields an identification of a suit-able expression in terms of the Γ’s, with a certain combinationof the L’s.

(3) The combination of the L’s is expressed in terms of Gaussiancurvature.

The first two steps were carried out in Proposition 10.8.2. Wechoose the valus i = j = 1 and k = q = 2 for the indices. ApplyingTheorem 9.9.1(c), we obtain

Γ21[1;2] + Γm1[1Γ

22]m = −L1[1L

22]

= g1iLi[1L

22]

= g11L1[1L

22]

since the term L2[1L

22] = 0 vanishes. This yields the desired formula

for K and completes the proof of the theorema egregium. �

CHAPTER 11

Signed curvature of plane curves

11.1. Curvature expressed in terms of θ(s)

We identify R2 with C so that a vector in the plane can be writtenas a complex number.

Let s be an arclength parameter along a curve α(s) with tangentvector v(s) = α′(s).

Definition 11.1.1 (function theta). The angle θ(s) is defined inone of the following two equivalent ways:

(1) We write v(s) = eiθ(s), where the angle θ(s) is measured coun-terclockwise, from the positive ray of the x-axis, to the vec-tor v(s).

(2) Using a suitable branch of the complex logarithm, we can alsoexpress θ(s) as follows: θ(s) = 1

ilog v(s) = −i log v(s).

Lemma 11.1.2. If α(s) = (x(s), y(s)) then dxds

= cos θ and dyds

=sin θ.

Proof. Indeed, v(s) = eiθ = cos θ + i sin θ. �

Lemma 11.1.3. We have the relation ddθeiθ = ieiθ.

This was proved in complex functions.

Definition 11.1.4. The signed curvature kα of the curve α is

kα(s) =dθ

ds. (11.1.1)

This is discussed in more detail in Section 11.7.

Theorem 11.1.5. We have∣∣∣kα∣∣∣ = kα.

Proof. Recall that v(s) = eiθ(s). By chain rule, we have

dv

ds= ieiθ(s)

dθ

ds,

and therefore

|kα(s)| =∣∣∣∣dθ

ds

∣∣∣∣ =∣∣∣∣dv

ds

∣∣∣∣ =∣∣∣∣d2α

ds2

∣∣∣∣ = kα(s)

113

114 11. SIGNED CURVATURE OF PLANE CURVES

as required. �

11.2. Signed curvature with respect to an arb. parameter

The formula for the curvature of a parametrized curve α(s) in Eu-clidean space is particularly simple with respect to the arclength param-eter s, namely kα(s) = |α′′(s)|; see formula (4.1.1). For plane curves,the curvature can be expressed in terms of an arbitrary parameter t ofa regular parametrisation, as follows.

Theorem 11.2.1. Let α(t) = (x(t), y(t)) be an arbitrary regularparametrisation (not necessarily arclength) of a plane curve. Then

kα =xy − yx

‖α‖3 . (11.2.1)

Proof. With respect to an arbitrary parameter t, the componentsof the tangent vector v(t) = α(t) are x(t) and y(t). Hence we have

tan θ =y

x(11.2.2)

so that, differentiating (11.2.2) with respect to t, we obtain

1

cos2 θ

dθ

dt=xy − xy

x2(11.2.3)

and therefore

dθ

dt=xy − xy

x2cos2 θ =

cos2 θ

x2(xy − xy) (11.2.4)

Meanwhile by chain rule and Lemma 11.1.2, we have

x =dx

dt=dx

ds

ds

dt= cos θ

ds

dtand therefore

cos θ

x=dt

ds.

Substituting into (11.2.4) we obtain

dθ

dt=

(dt

ds

)2

(xy − xy)

Furthermore, dθdt

= dθdsdsdt

and therefore

dθ

ds=xy − xy(dsdt

)3

Next, dsdt

= |v(t)| = |α(t)| = |dαdt|. Thus

dθ

ds=xy − yx

|α|3

11.4. GAUSS MAP 115

and by (11.1.1) we obtain the desired formula (11.2.1). �

Corollary 11.2.2. With respect to an arbirary regular parame-ter t, the formula for the curvature of a regular plane curve α(t) =(x(t), y(t)) is

kα(t) =

∣∣det [α α]∣∣

‖α‖3 . (11.2.5)

Here x = dx/dt, y = dy/dt, and α = dα/dt is the tangent vector tothe curve α(t) = (x(t), y(t)). Here we used the 2-by-2 matrix

[α α] ,

where α and α are written as column vectors, yielding the determi-nant det [α α] = xy − yx. This was proved in see Section 11.1; seeformula (11.2.1).

Example 11.2.3. Calculate the curvature of the graph of y = f(x)at a critical point c using formula (11.2.5).

11.3. Jordan curves in the plane

Recall that S1 is the circle.

Definition 11.3.1. A Jordan curve in the Euclidean plane is a non-selfintersecting closed curve that can be represented by a continuousone-to-one map

α : S1 → R2.

Theorem 11.3.2 (Jordan curve theorem). A Jordan curve sepa-rates the plane into two open regions: a bounded region and an un-bounded region.

The bounded region is called the interior region.We will only deal with smooth (i.e., infinitely differentiable) regular

(i.e, α 6= 0) maps α.

11.4. Gauss map

Consider a Jordan curve admitting a smooth parametrisation, de-noted C, in C:

C ⊆ C.

By Theorem 5.1.1 there is an arclength parametrisation α(s) of C,where s ranges through the interval [0, L] where L is the length of Cand α(0) = α(L). Here we assume that α′(0) = α′(L).


Definition 11.4.1. Let

v(s) = α′(s)

be the velocity vector of the parametrisation α at the point α(s) ∈ C,thought of as a point of the unit circle S1 ⊆ C

v(s) ∈ S1

by translating the initial point of the vector v(s) to the origin.

Remark 11.4.2. The Gauss map is usually defined using the nor-mal vector n(s) (see Remark 11.6.5). For plane curves the tangentvector v(s) and the normal vector differ by a 90◦ rotation in the plane:n(s) = iv(s) for an arclength parameter s so it could be defined interms of v as well.

The normal vector n(s) = iv(s) defines a map to the unit circlecalled the Gauss map, as follows.

Definition 11.4.3. The Gauss map of a smooth Jordan curve C ⊆C is the map

n : C → S1, α(s) 7→ n(s) = iv(s) (11.4.1)

where α(s) ∈ C and v(s) = α′(s).

Example 11.4.4. If C = S1 is the unit circle itself, then the Gaussmap is the identity map G : S1 → S1.

11.5. Convex Jordan curves

Recall that the set-theoretic complement of a subset B ⊆ A isdenoted A \B.

Recall that any line ℓ ⊆ R2 divides the plane into two open half-planes, namely the two connected components1 of the set comple-ment R2 \ ℓ.

Definition 11.5.1. A smooth Jordan curve C ⊆ R2 is calledstrictly convex 2 if one of the following two equivalent conditions is sat-isfied:

(1) the interior of each segment joining a pair of points of C iscontained in the interior region of C;

(2) consider the tangent line Tp to C at a point p ∈ C; then thecomplement C \ {p} lies entirely in one of the open halfplanesof R2 \ Tp defined by the tangent line, for all p ∈ C.

1rechivei kshirut2kamur with kuf

11.5. CONVEX JORDAN CURVES 117

Remark 11.5.2. A counterclockwise parametrisation correspondsto θ(s) increasing, while a clockwise parametrisation corresponds to θ(s)decreasing.

Theorem 11.5.3. Assume that a smooth regular Jordan curve Cis strictly convex, and parametrized counterclockwise. Then the Gaussmap n : C → S1 of (11.4.1) is one-to-one and onto, and the func-tion θ(s) is monotone increasing.

Proof of the “onto” part. We will work instead with the mapv : C → S1. This is legitimate since it only differs from n by a 90 degreerotation.

Let α(s) be an arclength parametrisation of C. First we considerthe case of “horizontal” vectors v(s) in the (x, y)-plane. These occurat points on C with maximal and minimal imaginary part, i.e., the y-coordinate.

We think of the y corodinate as defining a height function on thecurve. By applying Rolle’s theorem to the height function y(s), weobtain a point of minimum smin and a point of maximum smax.

The points α(smin) and α(smax) have “opposite” horizontal tangentvectors. Such points correspond to the values θ = 0 and θ = π.

To treat the general case, we will now show how one obtains thepair of opposite tangent vectors

v = eiθ and − v = ei(θ+π).

First consider the vector nθ = ei(θ+π2) normal to v. Consider the func-

tion f(s) given by the scalar product

f(s) = 〈α(s), nθ〉where α(s) is a parametrisation of the curve C. We seek the extremaof f . This is thought of as the replacement of the height function above.At an extremum s0 of the function f , we have

d

ds

∣∣∣∣s=s0

(f(s)) =

⟨dα

ds, nθ

⟩= 0.

Hence the tangent vector v(s0) = α′(s0) at each extremum s0 of f isparallel to v. As v ranges over S1, we thus obtain the points on thecurve where the tangent vector is parallel to v. �

Proof of the “one-to-one” part. We would like to show thatthe Gauss map (11.4.1) is one-to-one. Again we will work with tangentvectors in place of the normal vectors. We argue by contradiction.Suppose on the contrary that two distinct points p ∈ C and q ∈ Chave identical tangent vectors v(s) = eiθ. Then the tangent lines Tp


and Tq to C at p and q are parallel. By definition of convexity, thecurve C lies on the same side of each of the tangent lines Tp and Tq.Hence the tangent lines must coincide: Tp = Tq. Thus, both p and qmust lie on the common line Tp = Tq. This implies that the arc ofthe curve between p and q is contained in C. This contradicts thehypothesis that the curve is strictly convex. The contradiction provesthat the map is one-to-one. �

11.6. Total curvature of a convex Jordan curve

The result on the total curvature3 of a Jordan curve is of interest inits own right. Furthermore, the result serves to motivate an analogousstatement of the Gauss–Bonnet theorem for surfaces in Section 12.9.

Definition 11.6.1. The total curvature Tot(C) of a curve C witharclength parametrisation α(s) : [a, b] → R2 is the integral

Tot(C) =

∫ b

a

kα(s)ds. (11.6.1)

Example 11.6.2. Let us show that the total curvature of a circle Cr(of radius r) is 2π and therefore indepedent of r. Indeed, consider aparametrisation α(s) : [0, 2πr] → R2 of the circle given by the usualtrigonometric functions. Then

Tot(Cr) =

∫ 2πr

0

kα(s)ds =

∫ 2πr

0

1

rds = 2π.

Definition 11.6.3. If C is a closed curve, i.e, α(a) = α(b) andα′(a) = α′(b), we will write the integral (11.6.1) using the notation ofa contour integral

Tot(C) =

∮

C

kα(s)ds. (11.6.2)

Theorem 11.6.4. The total curvature of each strictly convex smoothJordan curve C with arclength parametrisation α(s) equals 2π, i.e.,

Tot(C) =

∮

C

kα(s)ds = 2π.

Proof. Let α(s) be a unit speed parametrisation so that α(0) bethe lowest point of the curve, and assume the curve is parametrizedcounterclockwise. Let v(s) = α′(s). Then v(0) = ei0 = 1 and θ(0) = 0.

3Akmumiyut kolelet

11.7. SIGNED CURVATURE kα OF JORDAN CURVES IN THE PLANE 119

The function θ = θ(s) is monotone increasing from 0 to 2π by The-orem 11.5.3. Applying the change of variable formula for integration,we obtain

Tot(C) =

∮

C

kα(s)ds =

∮

C

∣∣∣∣dv

ds

∣∣∣∣ ds =∮

C

dθ

dsds =

∫ 2π

0

dθ = 2π,


Remark 11.6.5. We can also consider the normal vector n(s) tothe curve. The normal vector satisfies n(s) = ei(θ+

π2) = iei(θ) and |dn

ds| =

|dvds| = dθ

ds. Therefore we can also calculate the total curvature as follows:∮

C

kα(s)ds =

∮

C

∣∣∣∣dn

ds

∣∣∣∣ ds =∮

C

dθ

dsds =

∫ 2π

0

dθ = 2π,

with n(s) in place of v(s). For surfaces the Gauss map is defined bymeans of the normal vector.

Example 11.6.6. The ellipse E is a closed convex curve. Recallthat an arbitrary ellipse defined by a quadratic equation ax2 + 2bxy +cy2+dx+ey = f , ac−b2 > 0, provided the curve is nondegenerate; seeDefinition 2.6.5 for details. Applying Theorem 11.6.4 we obtain that

Tot(E) = 2π.

Namely 2π is the total curvature of E.

Example 11.6.7. The hyperbola H defined by λ1x2 + λ2y

2 = k isnot a closed curve. In the special case λ1 = −λ2 (see Definition 2.6.2)when the asymptotes of H are orthogonal, the image of θ is preciselyhalf the circle. Therefore we can define the total curvature of this non-closed curve by a similar integral, and a similar integration argumentshows that the total curvature is π (rather than 2π).

Remark 11.6.8. A theorem similar to Theorem 11.6.4 in fact holdsfor an arbitrary regular Jordan curve (even though in general θ(s) willnot be a monotone function) provided we use signed curvature. Wehave dealt only with the convex case in order to simplify the topologicalconsiderations. See further in Section 11.7.

Remark 11.6.9. A similar calculation will yield the Gauss–Bonnettheorem for convex surfaces in Section 12.9.

11.7. Signed curvature kα of Jordan curves in the plane

Let α(s) be an arclength parametrisation of a Jordan curve in theplane. We assume that the curve is parametrized counterclockwise andlet v(s) = α′(s).


Remark 11.7.1. As in Section 11.4, a continuous branch of θ(s)can be chosen where θ(s) is the angle measured counterclockwise fromthe positive x-axis to the tangent vector v(s) = α′(s).

Remark 11.7.2. If the Jordan curve is not convex, the function θ(s)will not be monotone and at certain points its derivative may takenegative values:

θ′(s) < 0.

Once we have chosen a continuous branch of θ, we can define thesigned curvature kα of the parametrized Jordan curve as follows.

Definition 11.7.3. The signed curvature kα(s) of a plane Jor-dan curve α(s) oriented counterclockwise is defined globally by set-

ting kα(s) =dθds

where θ = 1ilog v, or equivalently v(s) = α′(s) = eiθ(s).

11.8. Rotation index of a closed curve in the plane

Consider a regular closed plane curve of length L with an arclengthparametrisation α(s) with α′(s) = eiθ(s). A continuous branch of θ(s)can be chosen even if the curve is not simple, obtaining a map θ : [0, L] →R. We can assume that θ(0) = 0. Then θ(L) is necessarily an integermultiple of 2π.

Definition 11.8.1. The rotation index ια of a closed unit speed

plane curve α(s) is the integer ια = θ(L)−θ(0)2π

.

Theorem 11.8.2. The rotation index of a smooth Jordan curveis ±1.

For a proof see See Millman & Parker [MP77, p. 55].

11.9. Total signed curvature

We have the following generalisation of Theorem 11.6.4 on the totalcurvature.

Definition 11.9.1. The total signed curvature of a smooth Jordancurve C of length L with arclength parametrisation α(s), s ∈ [0, L] isdefined to be

Tot(C) =

∫ L

0

kα(s)ds.

We obtain the following generalisation of Theorem 11.6.4.

11.9. TOTAL SIGNED CURVATURE 121

Figure 11.8.1. Three points on a curve with thesame tangent direction, but the middle one correspondsto θ′(s) < 0.

Theorem 11.9.2. The total signed curvature of a smooth Jordancurve C with counterclockwise arclength parametrisation α(s) is 2π:

Tot(C) =

∫ L

0

kα(s)ds = 2π.

Proof. We exploit the continuous branch θ(s) as in Theorem 11.6.4,but without the absolute value signs on the derivative of θ(s). Applyingthe change of variable formula for integration, we obtain

Tot(C) =

∮

C

kα(s)ds =

∮

C

dθ

dsds =

∫ 2π

0

dθ = 2π,

where the upper limit of integration is 2π by Theorem 11.8.2. �

11.9.1. Algebraic degree. This section is optional.

Definition 11.9.3. Let α(s) be a parametrisation of a smooth closedcurve C. Consider an arbitrary smooth map (not necessarily the Gaussmap) α(s) 7→ eiθ(s) from C to the circle. The algebraic degree of the mapat a point z ∈ S1 is defined to be the sum

∑θ−1(z) sign

(dθds

), where the

summation is over all points in the inverse image of z.

Here by Sard’s theorem z can be chosen in such a way that the inverseimage is finite so that the sum is well-defined.


Theorem 11.9.4. For Jordan curves (i.e., embedded curves), the alge-braic degree of the Gauss map is 1 if oriented counterclockwise and −1 iforientated counterclockwise.

For nonconvex Jordan curves, the Gauss map to the circle will not beone-to-one, but will still have an algebraic degree one. This phenomenonhas an analogue for embedded surfaces in R3 where the algebraic degree isproportional to the Euler characteristic.

Example 11.9.5. If the curve is not simple, the degree may be differentfrom ±1. Thus, the map defined by z 7→ zn restricted to the unit circle givesa map eiθ 7→ einθ of algebraic degree n. The preimage of 1 = ei0 consists

precisely of the nth roots of unity e2πikn , k = 0, 1, 2, . . . , n − 1. Altogether

there are n of them and therefore the algebraic degree is n.

11.9.2. Connected components of curves. This section is optional.Until now we have only considered connected curves. A curve may in generalhave several connected components.4

Definition 11.9.6. Two points p, q on a curve C ⊆ R2 are said to lie inthe same connected component if there exists a continuous map h : [0, 1] → Csuch that h(0) = p and h(1) = q.

This defines an equivalence relation on the curve C, and decomposes itas a disjoint union C =

⊔iCi.

Definition 11.9.7. The set of connected components of C is denoted π0(C).The number of connected components is denoted |π0(C)|.

Example 11.9.8. Let F (x, y) = (x2 + y2 − 1)((x − 10)2 + y2 − 1), andlet CF be the curve defined by F (x, y) = 0. Then CF is the union of a pair ofdisjoint circles. Therefore it has two connected components: |π0(CF )| = 2.

The total curvature can be similarly defined for a non-connected curve,by summing the integrals over each connected components.

Definition 11.9.9. The total curvature of a curve C =⊔iCi is Tot(C) =∑

iTot(Ci).

We have the following generalisation of the theorem on total curvatureof a curve.

Theorem 11.9.10. If each connected component of a curve C is a Jordancurve parametrized counterclockwise, then the signed total curvature of C

is Tot(C) = 2π |π0(C)|.Proof. We apply the previous theorem to each connected component,

and sum the resulting total curvatures. �

4Rechivei k’shirut.

CHAPTER 12

Surfaces, hyperbolic metric, Gauss–Bonnet

12.1. Gaussian curvature as product of signed curvatures

We originally defined the Gaussian curvature K = K(u1, u2) asthe determinant of the Weingarten map Wp at p = x(u1, u2) ∈ M .We also saw that the absolute value |K| can be represented as theproduct of the curvatures kβ1 and kβ2 of geodesics in the direction ofthe eigenvectors vi of Wp, namely |K| = kβ1kβ2.

Signed curvatures were defined in Section 11.1. Replacing thegeodesics by plane curves obtained by intersecting M with the planesspanned by n and each of the eigenvectors vi, as already mentioned inDefinition 9.6.1, item 9.6.3, for signed curvatures one has the sharperformula K = kβ1kβ2. In this spirit, some textbooks actually define theGaussian curvature not as the determinant of the Weingarten map butrather as the product of the signed curvatures kα of such a pair of planecurves.

Theorem 12.1.1. Let p ∈ M . Let v1, v2 be orthogonal eigenvec-tors of Wp : Tp → Tp. Let Ei = Span(vi, n) be the plane spanned byeigenvector vi and n. Consider the plane curve βi =M ∩Ei. Then theGaussian curvature Kp of M ⊆ R3 at p satisfies

Kp = kβ1kβ2

where kβi =dθids

is the signed curvature of the plane curve β1 and β2.

Remark 12.1.2. To explain this in terms of the Hessian, we can as-sume without loss of generality thatM is the graph of a function f(x, y)vanishing at the origin, with a critical point at the origin, and suchthat the eigenspaces of the Hessian of f are precisely the x-axis andthe y-axis. Then the Gaussian curvature at the origin is the product ofsigned curvatures of the curves obtained by restricting f respectivelyto the x-axis and the y-axis.

If we adjust the coordinates by an isometry in such a way that the xand y axes become the eigenspaces, while the z-axis is the direction ofthe normal, thenHf becomes a diagonal martixHf = diag(λ1, λ2). Thecurve β1 in the (x, z) plane is then described by z = ax2 + o(x2) while

123

124 12. SURFACES, HYPERBOLIC METRIC, GAUSS–BONNET

Weingartenmap (Lij)

Gaussian cur-vature K

mean cur-vature H

plane

(0 00 0

)0 0

cylinder

(1 00 0

)0 1

2

invarianceof curvature

yes no

Table 12.2.1. Plane and cylinder have the same intrinsicgeometry (K), but different extrinsic geometries (H)

the curve β2 in the (y, z) plane is described implicitly by z = cy2+o(y2).With respect to the natural orientations in these planes, the curves havesigned curvature respectively 2a and 2b at the origin, so that K = 4abas required.

12.2. Mean versus Gaussian curvature

An imporant consequence of the material of Chapter 10 is the fol-lowing.

Theorem 12.2.1. Unlike Gaussian curvature K, the mean curva-ture H = 1

2Lii cannot be expressed in terms of the metric coefficients gij

and their derivatives.

Proof. The plane and the cylinder have parametrisations withidentical (gij), but with different mean curvature, cf. Table 12.2.1. Tosummarize, Gaussian curvature is an intrinsic invariant, while meancurvature an extrinsic invariant, of the surface. �

12.3. The Laplace–Beltrami operator

A (u1, u2) chart in which the metric on M becomes conformal (seeDefinition 17.15.1) to the standard flat metric, is referred to as isother-mal coordinates . The existence of the latter is proved in [Bes87].

Definition 12.3.1. The Laplace–Beltrami operator for a metricon M of the form λ(u1, u2)δij , where λ > 0, in isothermal coordinatesis

∆LB =1

λ

(∂2

∂(u1)2+

∂2

∂(u2)2

).

12.4. LAPLACE-BELTRAMI FORMULA FOR GAUSSIAN CURVATURE 125

Γ1ij j = 1 j = 2

i = 1 µ1 µ2

i = 2 µ2 −µ1

Γ2ij j = 1 j = 2

i = 1 −µ2 µ1

i = 2 µ1 µ2

Table 12.4.1. Symbols Γkij of a metric e2µ(u1,u2)δij

The notation means that when we apply the operator ∆LB to afunction h = h(u1, u2), we obtain

∆LB(h) =1

λ

(∂2h

∂(u1)2+

∂2h

∂(u2)2

).

In more readable form for h = h(x, y), we have

∆LB(h) =1

λ

(∂2h

∂x2+∂2h

∂y2

).

In other words ∆LBh = 1λ∆0h where ∆0 is the flat Laplacian.

12.4. Laplace-Beltrami formula for Gaussian curvature

Theorem 12.4.1. Given a metric in isothermal coordinates withmetric coefficients gij = λ(u1, u2)δij, its Gaussian curvature is minushalf the Laplace-Beltrami operator applied to the ln of the conformalfactor λ:

K = −1

2∆LB(lnλ). (12.4.1)

Here ln x is the natural logarithm so that ddx

ln x = 1x.

Proof. The Gamma symbols were computed in Section 7.6 and aretabulated in Table 12.4.1, where λ = e2µ. We have from Table 12.4.1:

2Γ21[1;2] = Γ2

11;2 − Γ212;1 = −µ22 − µ11.

It remains to verify that the ΓΓ term in the expression (10.9.2) for theGaussian curvature vanishes:

2Γj1[1Γ22]j = 2Γ1

1[1Γ22]1 + 2Γ2

1[1Γ22]2

= Γ111Γ

221 − Γ1

12Γ211 + Γ2

11Γ222 − Γ2

12Γ212

= µ1µ1 − µ2(−µ2) + (−µ2)µ2 − µ1µ1

= 0.

Then from formula (10.9.2) we have

K =2

λΓ21[1;2] = −1

λ(µ11 + µ22) = −∆LBµ.


Meanwhile, ∆LB lnλ = ∆LB(2µ) = 2∆LB(µ), proving the result. �

Setting λ = f 2, we restate the theorem as follows.

Corollary 12.4.2. Given a metric in isothermal coordinates withmetric coefficients gij = f 2(u1, u2)δij, its Gaussian curvature is minusthe Laplace-Beltrami operator of the ln of the conformal factor f :

K = −∆LB ln f. (12.4.2)

12.5. Hyperbolic metric in the upperhalf plane

This example is also discussed in Section 17.25. Let x = u1 andy = u2.

Theorem 12.5.1. The metric with coefficients

gij =1

y2δij (12.5.1)

in the upperhalf plane

H2 = {(x, y) : y > 0} ,called the hyperbolic metric of the upper half plane, has constant Gauss-ian curvature K = −1.

Proof. By Theorem 12.4.2, we have

K = −∆LB ln f = ∆LB ln y = y2(− 1

y2

)= −1,

as required. �

Theorem 12.5.2. The differential equation for k = 1 of a hyper-bolic geodesic in the upperhalf plane is

x′′ − 2

yx′y′ = 0,

and for k = 2 it is

y′′ +1

y(x′)2 − 1

y(y′)2 = 0.

Proof. In Section 7.6 we computed the Gamma symbols of a met-ric relative to isothermal coordinates. We have Γ1

11 = λ12λ, Γ1

22 = −λ12λ

,

and Γ112 =

λ22λ. Since λ(x, y) = y−2 for the hyperbolic metric we obtain

Γ111 = Γ1

22 = 0 while

Γ112 =

λ22λ

=−2y−3

2y−2= −y

2

y3= −1

y. (12.5.2)

12.5. HYPERBOLIC METRIC IN THE UPPERHALF PLANE 127

Thus the differential equation of a geodesic αk′′+Γkijα

i′αj′= 0 for k = 1

becomes α1′′ + 2Γ112α

1′α2′ = 0. Since α1 = x and α2 = y we obtainx′′+2Γ1

12x′y′ = 0 which using formula (12.5.2) becomes x′′− 2

yx′y′ = 0.

Meanwhile Γ211 = −λ2

2λ= 1

yand similarly Γ2

22 = − 1ywhile Γ2

12 =λ12λ

= 0.

Therefore the geodesic equation for k = 2 is y′′+ 1yx′x′− 1

yy′y′ = 0. �

Example 12.5.3. The exponential function y = es (while x is con-stant as a function of s) satisfies both equations. Therefore it providesa parametrisation of a hyperbolic geodesic which traces out a verticalhalf-line in the upperhalf plane.

Remark 12.5.4. The pseudosphere is an example of a surface ofconstant Gaussian curvature −1 embedded in Euclidean space; see Sec-tion 5.11.

Either one of the formulas (10.9.2), (12.4.1), or (12.4.2) can serveas the intrinsic definition of Gaussian curvature, replacing the extrinsicdefinition (10.9.1), cf. Remark 10.6.

Remark 12.5.5. For a reader familiar with elements of Riemanniangeometry including Jacobi fields, it is worth mentioning that the Jacobiequation y′′ +Ky = 0 of a Jacobi field y on M (expressing an infini-tesimal variation by geodesics) sheds light on the nature of curvaturein a way that no mere formula for K could. Thus, in positive cur-vature, geodesics converge, while in negative curvature, they diverge.However, to prove the Jacobi equation, one would need to have alreadyan intrinsically well-defined quantity on the left hand side, y′′+Ky, ofthe Jacobi equation. In particular, one would need an already intrinsicnotion of curvature K. Thus, a proof of the theorema egregium nec-essarily precedes the deeper insights provided by the Jacobi equation.Similarly, the Gaussian curvature at p ∈M is the first significant termin the asymptotic expansion of the length of a “small” circle of center p.This fact, too, sheds much light on the nature of Gaussian curvature.However, to define what one means by a “small” circle, requires in-troducing higher order notions such as the exponential map, whichare usually understood at a later stage than the notion of Gaussiancurvature, cf. [Ca76, Car92, Ch93, GaHL04].

12.5.1. Binet–Cauchy identity. This section is optional.

Theorem 12.5.6 (Binet–Cauchy identity). The 3-dimensional case ofthe Binet–Cauchy identity is the identity (a · c)(b · d) = (a · d)(b · c) + (a ×b) · (c× d), where a, b, c, and d are vectors in R3.

The formula can also be written as a formula giving the dot product oftwo wedge products, namely (a× b) · (c× d) = (a · c)(b · d)− (a · d)(b · c).


Corollary 12.5.7 (Special case of Binet–Cauchy). In the special caseof vectors a = c and b = d, we obtain |a× b|2 = |a|2|b|2 − |a · b|2.

Example 12.5.8. When both vectors a, b are unit vectors, we obtain theusual relation 1 = cos2(φ) + sin2(φ) (here the vector product gives the sineand the scalar product gives the cosine) where φ is the angle between thevectors.

Corollary 12.5.9. Let a = x1 and b = x2 be the two tangent vectors tothe surface with parametrisation x(u1, u2). Then |x1×x2|2 = g11g22− g212 =det(gij).

Equivalently, we have |x1 × x2| =√

det(gij). This can be thought of as

the area of the parallelogram spanned by the two vectors.

12.6. Area elements of the surface, orientability

Consider a surface M embedded in 3-space. We have

√det(gij) =

∣∣x1 × x2∣∣ ,

where xi =∂x∂ui

(cf. the Binet–Cauchy identity 12.5.6).Generalizing the case of the area element rdrdθ in the plane in

polar coordinates as in Section 8.2, as well as the spherical element ofarea sinϕdθdϕ, we obtain the following definition.

Definition 12.6.1. The area element dAM of the surface M is

dAM =√

det(gij)du1du2 = |x1 × x2|du1du2 (12.6.1)

where the gij are the metric coefficients of the surface with respect tothe parametrisation x(u1, u2).

This notion of area is discussed in more detail in Section 17.14.We have used the subscript M so as to specify which surface we are

dealing with.

Definition 12.6.2. A surface M is orientable if it admits a con-tinuous unit normal vector field N = Np, defined at each point p ∈ M .

The vector field N is a globally defined field along an orientablesurface. Note that the image vector N can be thought of as an elementof the unit sphere: Np ∈ S2 ⊆ R3; see Section 12.7 for more details.

12.8. SPHERE S2 PARAMETRIZED BY GAUSS MAP OF M 129

12.7. Gauss map for surfaces in R3

Definition 12.7.1. The Gauss map of an orientable surface M ⊆R3, denoted

G : M → S2,

is the map sending each point p = x(u1, u2) to the normal vector G(p) =n(u1, u2).

Remark 12.7.2. Each point p ∈ M lies in a neighborhood witha suitable parametrisation x(u1, u2). At a point p = x(u1, u2), wehave a normal vector n(u1, u2), obtained by normalizing the vectorproduct x1 × x2, which coincides with the globally selected normal N ,i.e., n(u1, u2) = Nx(u1,u2).

Lemma 12.7.3. If K 6= 0 at p ∈ M then the tangent vectorsWp(x1),Wp(x2) ∈ Tp are linearly independent.

Proof. The coefficients of the matrix (Li j) the coordinates of∂n∂u1, ∂n∂u2

with respect to the basis (x1, x2). Hence independence of thevectors n1 and n2 is equivalent to the condition K = det(Li j) 6= 0. �

Theorem 12.7.4. Choose a point p = x(a, b) ∈ M of the surface.If the Gaussian curvature is nonzero at p, then the map n(u1, u2) from(a neighborhood of) M to S2 produces a regular parametrisation of aspherical neighborhood of the point

G(p) = n(a, b) ∈ S2

on the sphere, where G is the Gauss map.

Proof. The parametrisation is given by the map G to the sphere.We have

∂

∂ui(G(p)) =

∂

∂ui(n(u1, u2)) =

∂n

∂ui=Wp(xi).

By Lemma 12.7.3, the two partials ofG are linearly independent. Hencethe parametrisation defined by G is regular in a neighborhood of p. �

12.8. Sphere S2 parametrized by Gauss map of M

Remark 12.8.1. The vector n(u1, u2) was originally defined as anormal to the original surface M , but we now with to think of it asgiving a parametrisation of an open neighborhood on the unit sphere.Now consider the area element of the unit sphere.

Definition 12.8.2. Assume K 6= 0 at a point of M . Let (gαβ)be the metric coefficients of the regular parametrisation n(u1, u2) of a

neighborhood on the sphere, where nα = ∂n(u1,u2)∂uα

for each α = 1, 2.


Theorem 12.8.3. Consider a parametrisation n(u1, u2) of a neigh-borhood on the sphere S2 as in Theorem 12.7.4. Then the area ele-ment dAS2 can be expressed as

dAS2 =√

det(gαβ)du1du2 = |n1 × n2|du1du2.

Proof. This follows from the usual formula for the element ofarea for surfaces, applied to the chosen parametrisation n(u1, u2) asdefined in Theorem 12.7.4, in place of the traditional parametrisationin spherical coordinates (cf. Binet–Cauchy in Section 12.5.1). �

Remark 12.8.4. We have used the subscript S2 to distinguish thearea element dAS2 of the sphere S2 from the area element dAM of thesurfaceM as in (12.6.1). Note that we need to distinguish the two areaelements

dAS2 and dAM

because both will occur in the proof of the Gauss–Bonnet theorem.

12.9. Comparison of two parametrisations

When the Gaussian curvature is nonzero, the normal vector of Mallows us to define local coordinates n(u1, u2) on the unit sphere S2 asin Section 12.7.

Lemma 12.9.1. Consider the metric coefficients (gij) of the paramet-risation x(u1, u2) of the surface M and the metric coefficients (gαβ) ofthe sphere S2 relative to the parametrisation n(u1, u2). Then we havethe identity

det(gαβ) =(K(u1, u2)

)2det(gij)

where K = K(u1, u2) is the Gaussian curvature of the surface M .

Proof. Let L = (Lij) be the matrix of the Weingarten map W =

Wp with respect to the basis (x1, x2). Here p = x(u1, u2). By definitionof curvature we have K = det(L). Recall that the coefficients Lij ofthe Weingarten map are defined by

nα = Liαxi = xiLiα. (12.9.1)

Consider the 3 × 2-matrices A = [x1 x2] and B = [n1 n2]. Thenformula (12.9.1) implies by definition of matrix multiplication that

B = AL.

Therefore the Gram matrices satisfy

Gram(n1, n2) = BtB = (AL)tAL = LtAtAL = Lt Gram(x1, x2) L.

12.9. COMPARISON OF TWO PARAMETRISATIONS 131

Applying the determinant, we obtain

det (Gram(n1, n2)) = det (Gram(x1, x2)) det2(L),

and complete the proof of the lemma since det(L) = K. �

Remark 12.9.2. By the Cauchy-Binet formula, the desired identityis equivalent to the formula |nu1 × nu2 | = |det(Lij)| |xu1 × xu2 |, imme-diate from the observation that a linear map multiplies the area ofparallelograms by its determinant. Namely, the Weingarten map sendseach vector xi to ni: Wp(xi) = ni ∈ Tp. A stronger identity is truethat is sensitive to the sign of the Gaussian curvature: W (u)×W (v) =det(W )(u× v) where (u, v) is any basis of the tangent plane; e.g., thebasis (x1, x2).

CHAPTER 13

Gauss–Bonnet theorem

13.1. Special case of Gauss–Bonnet

The Gauss–Bonnet theorem for surfaces is to a certain extent anal-ogous to the theorem on the total curvature of a plane curve (Theo-rem 11.6.4). In both cases, an integral of curvature turns out to havetopological significance. We will only treat a special case of Gauss–Bonnet.

Theorem 13.1.1 (Special case of Gauss–Bonnet). Let M be anorientable closed surface in R3 of positive Gaussian curvature (a typicalexample is an ellipsoid). Then the curvature integral satisfies

∫

M

Kp dAM = 4π,

where Kp is the Gaussian curvature of M at the point p.

Remark 13.1.2. Note that 4π = 2π χ(S2) where χ is the Eulercharacteristic (see Section 13.4).

To prove Theorem 13.1.1, we will use the existence of global coor-dinates (with singularities only at the north and south poles) on thesphere to write down a concrete version of the calculation. We recallthe following.

(1) We consider a regular parametrisation x(u1, u2) of a surfaceM ⊆ R3.

(2) At every point ofM where the parametrisation x is defined, wehave the normal vector n(u1, u2) so that (x1, x2, n) is a basisfor R3.

(3) IfM is an orientable surface, a continuous Gauss mapG : M →S2 is defined by the normal vector n(u1, u2) as in Section 12.7.

(4) (θ, ϕ) are the usual spherical coordinates on the unit sphere.

We now consider the usual parametrisation

σ = σ(θ, φ) : [0, 2π)× (0, π) → S2

of the sphere as a surface of revolution (omitting the problematic poles),as in Section 5.6. Thus σ(θ, ϕ) = (sinϕ cos θ, sinϕ sin θ, cosϕ) ∈ S2.

133

134 13. GAUSS–BONNET THEOREM

Definition 13.1.3. Consider the parametrisation x(θ, ϕ) ofM givenby the composition x = G−1 ◦σ, and denote by (gij) the correspondingmetric coefficients.

Draw figure for the composition G−1 ◦ σ.

13.2. Proof of Gauss–Bonnet Theorem

We can dispense with local coordinate charts and instead computethe Gauss–Bonnet integral relative to coordinates (θ, ϕ) as in Defini-tion 13.1.3 (because they only omit two points). Let (gij) be the metricof M with respect to the parametrisation of Definition 13.1.3. Thenthe element of area can be written as

dAM =√det(gij)dθdϕ.

We will exploit the relation K(θ, ϕ)√det(gij) =

√det(gαβ) from Sec-

tion 12.9, where u1 = θ and u2 = ϕ. Here gαβ are the metric coefficientsof the standard metric on S2 with respect to parametrisation definedby the Gauss map G of M . We obtain

∫

M

K(θ, ϕ) dAM =

∫∫K(θ, ϕ)

√det(gij)dθdϕ

=

∫∫ √det(gαβ)dθdϕ

=

∫∫dAS2

= area(S2) = 4π,

as required.

Remark 13.2.1. Each of the double integrals∫∫

is the iterated

integral∫ π0

(∫ 2π

0(. . .)dθ

)dϕ, where the appropriate integrand is placed

where the dots (. . .) are.

13.2.1. Alternative proof. This material is optional. Here we present

another proof of Gauss–Bonnet theorem. The convexity of the surface

guarantees that the map n is one-to-one (compare with the proof of Theo-

rem 11.5.3 on closed curves). The integrand K dAM in a coordinate chart

(u1, u2) can be written as K(u1, u2) dAM . By Lemma 12.9.1, we have

K(u1, u2) dAM = K(u1, u2)√

det(gij)du1du2 =

√det(gαβ)du

1du2 = dAS2 .

Thus, the expression K dAM coincides with the area element dAS2 of the

unit sphere S2 in every coordinate chart. Hence we can write∫M K dAM =∫

S2 dAS2 = 4π, proving the theorem.

13.4. EULER CHARACTERISTIC 135

13.3. Local Gauss–Bonnet

There are various generalisations of the above theorem. One ofthem is the following local version of the theorem.

Theorem 13.3.1 (Local version of Gauss–Bonnet theorem). Con-sider a geodesic triangle (homeomorphic to a disk) T ⊆ M with an-gles α, β, γ. Then the integral of the Gaussian curvature over T is theangular excess: ∫

T

KdA = α + β + γ − π.

Corollary 13.3.2. The area of a spherical triangle with angles α,β, γ is the angular excess α + β + γ − π.

Proof. We apply the local Gauss–Bonnet theorem and notice thatfor M = S2, we have K = 1 at every point. �

13.4. Euler characteristic

Definition 13.4.1. Assume a surface M is partitioned into trian-gles. Then the Euler characteristic χ(M) of M is

χ(M) = V − E + F,

where V,E, F are respectively the numbers of vertices, edges, and faces(i.e., triangles) of M .

The Euler characteristic of a closed embedded surface in Euclidean3-space can be computed via the integral of the Gaussian curvature. Itcan be thought of as a generalisation of the rotation index of a planeclosed curve.

Theorem 13.4.2 (Gauss–Bonnet). The Euler characteristic χ(M)of a surface M satisfies∫

M

KpdAM = 2π χ(M), (13.4.1)

where in a coordinate patch Kp = K(u1, u2) is the Gaussian curvatureat the point p = x(u1, u2) ∈M .

Here we no longer assume that the curvature is positive.

Remark 13.4.3. The relation (13.4.1) is similar to the line integralexpression for the rotation index in formula (13.5.1).

One way of proving Gauss–Bonnet for embedded surfaces is to usethe notion of algebraic degree for maps between surfaces, similar to thealgebraic degree of a self-map of a circle.


Remark 13.4.4. The Gauss–Bonnet theorem holds for all surfaces,whether orientable or not.

Example 13.4.5. The Euler characteristic of the real projectiveplane RP2 is 1. Therefore for every metric on RP2 we have

∫RP2 KdA =

2π.

Example 13.4.6. The Euler characteristic of the torus T 2 is 0.Therefore for every metric on T 2 we have

∫T 2 KdA = 0.

13.5. Rotation index of closed curves and total curvature

Theorem 13.5.1. Let α(s) be an arclength parametrisation of ageometric curve C. Then the rotation index ια is related to the total

signed curvature Tot(C) as follows:

Tot(C) = 2πια. (13.5.1)

Proof. The proof is the same as in Section 11.7. We exploit thecontinuous branch θ(s) as in Theorem 11.6.4. Applying the change ofvariable formula for integration, we obtain

Tot(C) =

∮

C

kα(s)ds =

∮

C

dθ

dsds =

∫ 2πια

0

dθ = 2πια,

where the upper limit of integration is 2πια by definition of the rotationindex. �

Remark 13.5.2. An analogous relation between the Euler charac-teristic of a surface and its total Gaussian curvature appears in Sec-tion 13.4.

Example 13.5.3. The rotation index of a figure-8 curve (known asthe lemniscate of Gerono, one of the Lissajous curves) is 0. Hence itstotal signed curvature vanishes.

Example 13.5.4. Describe an immersed curve with rotation in-dex 2.

13.6. Surfaces of revolution and isothermalisation

Below we will express the metric of a surface of revolution in isother-mal coordinates. Recall that if a surface is obtained by revolving acurve (r(φ), z(φ)), we obtain metric coefficients g11 = r2 and g22 =(drdφ

)2+(dzdφ

)2. Thus we obtain the following theorem.

13.6. SURFACES OF REVOLUTION AND ISOTHERMALISATION 137

Theorem 13.6.1. We have the following expression of the metricof surface of revolution in coordinates u1 = r, u2 = φ:

g11 = r2, g22 =(drdφ

)2+(dzdφ

)2, g12 = 0. (13.6.1)

We will use this theorem to carry out explicit isothermalisation inthe case of a surface of revolution.

Proposition 13.6.2. Let (r(φ), z(φ)), where r(φ) > 0, be an ar-clength parametrisation of the generating curve of a surface of revolu-tion M . Then the change of variable

ψ =

∫1

r(φ)dφ,

produces an isothermal parametrisation ofM in terms of variables (θ, ψ).With respect to the new coordinates, the first fundamental form is givenby a scalar matrix with coefficients r(φ(ψ))2δij.

The existence of such a parametrisation is predicted by the uni-formisation theorem (Theorem 14.9.2) in the case of a general surface.

Proof. Consider an arbitrary change of parameter φ = φ(ψ). Bychain rule,

dr

dψ=dr

dφ

dφ

dψ.

Now consider again the first fundamental form (13.6.1). We need toimpose the condition

g11 = g22to ensure the condition that the matrix of metric coefficients is a scalarmatrix. By Theorem 13.6.1, this is equivalent to the equation

r2 =

(dr

dψ

)2

+

(dz

dψ

)2

, (13.6.2)

giving the common value of g11 = g22. By chain rule, equation (13.6.2)is equivalent to

r2 =

((dr

dφ

)2

+

(dz

dφ

)2)(

dφ

dψ

)2

. (13.6.3)

In the case when the generating curve is parametrized by arclength,equation (13.6.3) becomes

r =dφ

dψ.

Solving for ψ, we obtain ψ =∫

dφr(φ)

. Substituting the new variable φ(ψ)

in place of φ in the parametrisation of the surface, we obtain a new


parametrisation of the surface of revolution in coordinates (θ, ψ), suchthat the first fundamental form is scalar with conformal factor λ =r2(φ(ψ)) as per equation (13.6.2). �

13.7. Curvature of pseudosphere

As an application of the isothermalisation of Section 13.6, we cal-culate the curvature of the pseudosphere. The pseudosphere (see Sec-tion 5.11) is the surface of revolution generated by functions r(φ) = eφ

and z(φ) =∫ φ0

√1− e2τdτ = −

∫ 0

φ

√1− e2τdτ , where −∞ < φ ≤ 0.

Its metric coefficients are given by (gij) =

(e2φ 00 1

).

Theorem 13.7.1. The pseudosphere has constant Gaussian curva-ture K = −1.

Proof. Instead of applying the general formula for curvature, weuse the trick of a change of coordinates that results in isothermal co-ordinates so we can apply the formula for curvature in terms of theLaplace–Beltrami operator. We apply Proposition 13.6.2 to introducethe change of coordinates

ψ =

∫dφ

r(φ)=

∫e−φdφ = −e−φ. (13.7.1)

This results in isothermal coordinates (θ, ψ) with conformal factorf(ψ) = r(φ(ψ)). From (13.7.1) we obtain φ = − ln(−ψ) and there-fore

f(ψ) = r(φ(ψ)) = e− ln(−ψ) =1

eln(−ψ)=

1

−ψ = (−ψ)−1.

We have λ = f 2 = 1ψ2 . Recall that the metric with gij(θ, ψ) =

1ψ2 δij is

by definition hyperbolic. Therefore it has curvature K = −1 as shownin Section 12.5. For completeness we carry out the calculation as fol-lows. In isothermal coordinates (θ, ψ), Gaussian curvature is given bythe formula in terms of the Laplace–Beltrami operator. The Gaussiancurvature of the pseudosphere is

K = −∆LB ln f

= −1

λ

∂2

∂ψ2ln f(ψ)

= −ψ2 ∂2

∂ψ2ln((−ψ)−1)

= ψ2 ∂2

∂ψ2ln(−ψ).

13.7. CURVATURE OF PSEUDOSPHERE 139

Differentiating with respect to ψ, we obtain

K = ψ2 ∂

∂ψ

(−1

−ψ

)

= ψ2 ∂

∂ψ

(1

ψ

)

= ψ2

(− 1

ψ2

)

= −1,

as required. �

CHAPTER 14

Duality in linear algebra, calculus, diff. geometry

14.1. Transition from classical to modern diff geom

Remark 14.1.1. The theorema egregium of Gauss marks the tran-sition from classical differential geometry of curves and surfaces em-bedded in 3-space, to modern differential geometry of surfaces (andmanifolds) studied intrinsically.

More specifically, once we have a notion of Gaussian curvature (andmore generally sectional curvature) that only depends on the metricon a surface (or manifold), we can study the geometry of the surface(or manifold) intrinsically, i.e., without any reference to a Euclideanembedding.

Remark 14.1.2. To formulate the intrinsic viewpoint, one needsthe notion of duality of vector and covector. This theme is treated inChapter 14.

14.2. Duality in linear algebra; 1-forms

Let V be a finite-dimensional real vector space.

Example 14.2.1. Euclidean space Rn is an example of a real vectorspace of dimension n with basis (e1, . . . , en).

Example 14.2.2. The tangent plane TpM of a regular surface Mat a point p ∈ M is a real vector space of dimension 2.

Definition 14.2.3. A linear form, also called 1-form, φ on V is alinear functional

φ : V → R.

Definition 14.2.4 (dx, dy). In the usual Euclidean plane of vectors

v = v1e1 + v2e2

represented by arrows, we denote by dx the 1-form which extracts theabscissa of the vector, and by dy the 1-form which extracts the ordinateof the vector:

dx(v) = v1,

141

142 14. DUALITY IN LINEAR ALGEBRA, CALCULUS, DIFF. GEOMETRY

anddy(v) = v2.

Example 14.2.5. For a vector v = 3e1+4e2 with components (3, 4)we obtain

dx(v) = 3, dy(v) = 4.

Definition 14.2.6. The quadratic forms dx2 and dy2 are definedby squaring the value of the 1-form on v:

dx2(v) = (dx(v))2.

Such quadratic forms are called rank-1 quadratic forms.

Thus,dx2(v) = (v1)2, dy2(v) = (v2)2.

Example 14.2.7. In the case v = 3e1 + 4e2 we obtain dx2(v) = 9and dy2(v) = 16.

14.3. Dual vector space

Definition 14.3.1. The dual space of V , denoted V ∗, is the spaceof all linear forms on V :

V ∗ = {φ : φ is a 1-form on V } .Evaluating φ at an element x ∈ V produces a scalar φ(x) ∈ R.

Definition 14.3.2. The evaluation map1 is the natural pairing be-tween V and V ∗, namely a linear map denoted

〈 , 〉 : V × V ∗ → R,

defined by evaluating y at x, i.e., setting 〈x, y〉 = y(x), for all x ∈ Vand y ∈ V ∗.

Remark 14.3.3. Note we are using the same notation for the pair-ing as for the scalar product in Euclidean space. The notation is quitewidespread.

Definition 14.3.4. If V admits a basis of vectors

(xi) = (x1, x2, . . . , xn),

then the dual space V ∗ admits a unique basis, called the dual ba-sis (yj) = (y1, . . . , yn), satisfying

〈xi, yj〉 = δij, (14.3.1)

for all i, j = 1, . . . , n, where δij is the Kronecker delta function.

1hatzava?

14.4. DERIVATIONS 143

Example 14.3.5. Let V = R2. We have the standard basis e1, e2for V . The 1-forms dx, dy form a basis for the dual space V ∗. Thenthe basis (dx, dy) is the dual basis to the basis (e1, e2).

Example 14.3.6. Let V = R2 identified with C for convenience.We have the basis (1, e

iπ3 ) for V where 1 = e1 + 0e2 while

eiπ3 =

1

2e1 +

√3

2e2.

Find the dual basis in V ∗. The answer is y1 = dx− 1√3dy, y2 =

2√3dy.

14.4. Derivations

Let E be a space of dimension n, and let p ∈ E be a fixed point.The following definition is independent of coordinates.

Definition 14.4.1. Let

Dp = {f : f ∈ C∞}be the ring of real C∞-functions f defined in an arbitrarily small openneighborhood of p ∈ E.

Remark 14.4.2. The ring operations are pointwise multiplicationand pointwise addition, where we choose the intersection of the twodomains as the domain of the new function.

Note that Dp is infinite-dimensional as it includes all polynomials.

Theorem 14.4.3. Choose coordinates (u1, . . . , un) in E. A partialderivative ∂

∂uiat p is a 1-form

∂

∂ui: Dp → R

on the space Dp, satisfying Leibniz’s rule

∂(fg)

∂ui

∣∣∣∣p

=∂f

∂ui

∣∣∣∣p

g(p) + f(p)∂g

∂ui

∣∣∣∣p

(14.4.1)

for all f, g ∈ Dp.

Formula (14.4.1) can be written briefly as

∂∂ui

(fg) = ∂∂ui

(f)g + f ∂∂ui

(g),

with the understanding that both sides are evaluated at the point p.This was proved in calculus. Formula (14.4.1) motivates the followingmore general definition of a derivation.


Definition 14.4.4. A derivation X at the point p ∈ E is a 1-form

X : Dp → R

on the space Dp satisfying Leibniz’s rule:

X(fg) = X(f)g(p) + f(p)X(g) (14.4.2)

for all f, g ∈ Dp.

Remark 14.4.5. Linearity of a derivation is required only withregard to scalars in R, not with respect to functions.

14.5. Characterisation of derivations

It turns out that the space of derivations is spanned by partialderivatives.

Proposition 14.5.1. Let E be an n-dimensional space, and p ∈E. Then the collection of all derivations at p is a vector space ofdimension n.

Proof in case n = 1. We will first prove the result in the case n =1 of a single variable u at the point p = 0. Let X : Dp → R be aderivation. Then X(1) = X(1 · 1) = 2X(1) by Leibniz’s rule. There-fore X(1) = 0, and similarly for any constant by linearity of X .

Now consider the monic polynomial u = u1 of degree 1, viewedas a linear function u ∈ Dp=0. We evaluate the derivation X at theelement u ∈ D and set c = X(u). By the Taylor remainder formula, anyfunction f ∈ Dp=0 can be written as f(u) = a + bu + g(u)u where b =∂f∂u(0) and g is smooth and g(0) = 0. Now we have by linearity

X(f) = X(a+ bu+ g(u)u)

= bX(u) +X(g)u(0) + g(0) ·X(u)

= bc+ 0 + 0

= c∂

∂u(f).

Thus the derivation X coincides with the derivation c ∂∂u

for all f ∈Dp. Hence the tangent space is 1-dimensional and spanned by theelement ∂

∂u, proving the theorem in this case. �

14.5.1. The material in this subsection is optional. Proof in

case n = 2. Let us prove the result in the case n = 2 of two variables u, v atthe origin p = (0, 0). Let X be a derivation. Then X(1) = X(1 · 1) = 2X(1)by Leibniz’s rule. Therefore X(1) = 0, and similarly for any constant bylinearity of X. Now consider the monic polynomial u = u1 of degree 1,i.e., the linear function u ∈ Dp=0. We evaluate the derivation X at u and

14.6. TANGENT SPACE AND COTANGENT SPACE 145

set c = X(u). Similarly, consider the monic polynomial v = v1 of degree 1,i.e., the linear function v ∈ Dp=0. We evaluate the derivation X at v andset c = X(v). By the Taylor remainder formula, any function f ∈ Dp=0,

where f = f(u, v), can be written as f(u, v) = a + bu + bv + g(u, v)u2 +h(u, v)uv + k(u, v)v2, where the functions g(u, v), h(u, v), and k(u, v) are

smooth. Note that the coefficients b and b are the first partial derivativesof f at the origin. Now we have

X(f) = X(a+ bu+ bv + g(u, v)u2 + h(u, v)uv + k(u, v)v2)

= bX(u) + bX(v) +(X(g)u2 + g(u, v)2uX(u) + · · ·

)

= bX(u) + bX(v)

= bc+ bc

= c∂

∂u(f) + c

∂

∂v(f)

by evaluating at the point (0, 0). Thus the derivation X coincides with the

derivation c ∂∂u+c∂∂v for all test functions f ∈ Dp. Hence the two partials span

the tangent space. Therefore the tangent space is 2-dimensional, proving the

theorem in this case.

14.6. Tangent space and cotangent space

Definition 14.6.1. Let p ∈ E. The space of derivations at p iscalled the tangent space Tp = TpE at p.

The results of the previous section can be formulated as follows.

Corollary 14.6.2. Let (u1, . . . , un) be coordinates for E, and let p ∈E. Then a basis for the tangent space Tp is given by the n partialderivatives (

∂

∂ui

), i = 1, . . . , n.

Definition 14.6.3. The space dual to the tangent space Tp is calledthe cotangent space, and denoted T ∗

p .

Definition 14.6.4. An element of a tangent space is a vector, whilean element of a cotangent space is called a 1-form, or a covector.

Definition 14.6.5. The basis dual to the basis(∂∂ui

)is denoted

(duj), j = 1, . . . , n.

Thus each duj is by definition a linear form, or cotangent vector(covector for short). We are therefore working with dual bases

(∂∂ui

)


for vectors, and (duj) for covectors. The evaluation map as in (14.3.1)gives ⟨

∂

∂ui, duj

⟩= duj

(∂

∂ui

)= δji , (14.6.1)

where δji is the Kronecker delta.

14.7. Constructing bilinear forms out of 1-forms

Recall that the polarisation formula (see Definition 1.4.3) allowsone to reconstruct a symmetric bilinear form B = B(v, w), from thequadratic form Q(v) = B(v, v), at least if the characteristic is not 2:

B(v, w) =1

4(Q(v + w)−Q(v − w)). (14.7.1)

Similarly, one can construct bilinear forms out of the 1-forms dui,as follows.

Example 14.7.1. Consider a quadratic form ai(dui)2 defined by a

linear combination of the rank-1 quadratic forms (dui)2, as in Defini-tion 14.2.6.

Polarizing the quadratic form, one obtains a bilinear form on thetangent space Tp.

Example 14.7.2. Let v = v1e1+ v2e2 be an arbitrary vector in theplane. Let dx and dy be the standard covectors, extracting, respec-tively, the first and second coordinates of v. Consider the quadraticform Q given by Q = Edx2 + Fdy2, where E, F ∈ R. Here Q(v) iscalculated as

Q(v) = E(dx(v))2 + F (dy(v))2 = E(v1)2 + F (v2)2.

Polarisation then produces the bilinear form B = B(v, w), where vand w are arbitrary vectors, given by the formula

B(v, w) = E dx(v) dx(w) + F dy(v) dy(w).

Example 14.7.3. Setting E = F = 1 in the previous example, weobtain the standard scalar product in the plane:

B(v, w) = v · w = dx(v) dx(w) + dy(v) dy(w) = v1w1 + v2w2.

14.8. First fundamental form

Definition 14.8.1. A metric (or first fundamental form) g is asymmetric bilinear form on the tangent space at p, namely g : Tp×Tp →R, defined for all p and varying continuously and smoothly in p.

14.9. DUAL BASES IN DIFFERENTIAL GEOMETRY 147

Remark 14.8.2. In Riemannian geometry one requires the associ-ated quadratic form to be positive definite. In relativity theory oneuses a form of type (3, 1).

Recall that the basis for Tp in coordinates (ui) is given by thetangent vectors ∂

∂ui. These are given by certain derivations (see Sec-

tion 14.4).The first fundamental form g is traditionally expressed by a matrix

of coefficients called metric coefficients gij , giving the inner product ofthe i-th and the j-th vector in the basis: gij = g

(∂∂ui, ∂∂uj

), where g is

the first fundamental form.

Definition 14.8.3. The square norm the i-th vector is given by

the coefficient gii =∥∥ ∂∂ui

∥∥2.We will express the first fundamental form in more intrinsic notation

of quadratic forms built from 1-forms (covectors).

14.9. Dual bases in differential geometry

Let us now restrict attention to the case of 2 dimensions, i.e., thecase of surfaces. At every point p = (u1, u2), we have the metric coef-ficients gij = gij(u

1, u2). Each metric coefficient is thus a function oftwo variables.

We will only consider the case when the matrix is diagonal. This canalways be achieved, in two dimensions, at a point by a suitable changeof coordinates, by the uniformisation theorem (see Theorem 14.9.2).We set x = u1 and y = u2 to simplify notation. In the notationdeveloped in Section 14.7, we can write the quadratic form associatedwith the first fundamental form as follows:

g = g11(x, y)(dx)2 + g22(x, y)(dy)

2. (14.9.1)

For example, if the metric coefficients form an identity matrix: gij =δij, we obtain the standard flat metric

g = (dx)2 + (dy)2 (14.9.2)

or simply g = dx2 + dy2.

Example 14.9.1 (Hyperbolic metric). Let g11 = g22 = 1y2

at each

point (x, y) where y > 0. This means that∣∣ ∂∂u1

∣∣ = 1yand

∣∣ ∂∂u2

∣∣ = 1y.

The resulting hyperbolic metric in the upperhalf plane {y > 0} is ex-pressed by the quadratic form 1

y2(dx2 + dy2). Note that this expression

is undefined whenever y = 0. See Section 12.5. The hyperbolic metricin the upper half plane is a complete metric.


Closely related results are the Riemann mapping theorem and theconformal representation theorem.

Theorem 14.9.2 (Riemann mapping/uniformisation). Every met-ric on a connected2 surface is conformally equivalent to a metric ofconstant Gaussian curvature.

From the complex analytic viewpoint, the uniformisation theoremstates that every Riemann surface is covered by either the sphere, theplane, or the upper halfplane. Thus no notion of curvature is needed forthe statement of the uniformisation theorem. However, from the dif-ferential geometric point of view, what is relevant is that every confor-mal class of metrics contains a metric of constant Gaussian curvature.See [Ab81] for a lively account of the history of the uniformisationtheorem.

14.10. More on dual bases

Recall that if (x1, . . . , xn) is a basis for a vector space V then thedual vector space V ∗ possesses a basis called the dual basis and de-noted (y1, . . . , yn) satisfying 〈xi, yj〉 = yj(xi) = δij.

Example 14.10.1. In R2 we have a basis (x1, x2) =(∂∂x, ∂∂y

)in the

tangent plane Tp at a point p. The dual basis of 1-forms (y1, y2) for T∗p

is denoted (dx, dy). Thus we have⟨∂

∂x, dx

⟩= dx

(∂

∂x

)= 1

and ⟨∂

∂y, dy

⟩= dy

(∂

∂y

)= 1,

while ⟨∂

∂x, dy

⟩= dy

(∂

∂x

)= 0,

etcetera.

Similarly, in polar coordinates at a point p 6= 0 we have a basis(∂∂r, ∂∂θ

)for Tp, and a dual basis (dr, dθ) for T ∗

p . Thus we have⟨∂

∂r, dr

⟩= dr

(∂

∂r

)= 1

2kashir

14.10. MORE ON DUAL BASES 149

and ⟨∂

∂θ, dθ

⟩= dθ

(∂

∂θ

)= 1,

while ⟨∂

∂r, dθ

⟩= dθ

(∂

∂r

)= 0,

etcetera.Now in polar coordinates we have a natural area element r dr dθ.

Area of a region D is calculated by Fubini’s theorem as∫∫

D

r dr dθ =

∫ (∫rdr

)dθ.

Thus we have a natural basis (y1, y2) = (rdr, dθ) in T ∗p when p 6= 0,

i.e., y1 = rdr while y2 = dθ. Its dual basis (x1, x2) in Tp can be easilyidentified. It is

(x1, x2) =

(1

r

∂

∂r,∂

∂θ

).

Indeed, we have

〈x1, y1〉 =⟨1

r

∂

∂r, rdr

⟩= rdr

(1

r

∂

∂r

)= r

1

rdr

(∂

∂r

)= 1,

etcetera.

CHAPTER 15

Lattices and tori

15.1. Circle via the exponential map

We will discuss lattices in Euclidean space Rb in Section 15.2. Herewe give an intuitive introduction in the simplest case b = 1. Everylattice (discrete1 subgroup; see definition below in Section 15.2) in R =R1 is of the form

Lα = αZ = {nα : n ∈ Z} ⊆ R,

for some real α > 0. It is spanned by the vector αe1 (or −αe1).The lattice Lα ⊆ R is in fact an additive subgroup. Therefore

we can form the quotient group R/Lα. This quotient is a circle (seeTheorem 15.1.1). The 1-volume, i.e. the length, of the quotient circleis precisely α. We will give a description in terms of the complexfunction ez.

Theorem 15.1.1. The quotient group R/Lα is isomorphic to thecircle S1 ⊆ C.

Proof. Consider the map φ : R → C defined by

φ(x) = ei2πxα .

By the usual addition rule for the exponential function, this map is ahomomorphism from the additive structure on R to the multiplicativestructure in the group C \ {0}. Namely, we have

(∀x ∈ R)(∀y ∈ R) φ(x+ y) = φ(x)φ(y).

Furthermore, we have φ(x + αm) = φ(x) for all m ∈ Z and there-

fore ker φ = Lα. By the group-theoretic isomorphism theorem, themap φ descends to a map

φ : R/Lα → C,

which is injective. Its image is the unit circle S1 ⊆ C, which is a groupunder multiplication. �

1b’didah

151

152 15. LATTICES AND TORI

15.2. Lattice, fundamental domain

Let b > 0 be an integer.

Definition 15.2.1. A lattice L ⊆ Rb is the integer span of a linearlyindependent set of b vectors.

Thus, if vectors v1, . . . , vb are linearly independent, then they spana lattice

L = {n1v1 + . . .+ nbvb : ni ∈ Z} = Zv1 + Zv2 · · ·+ Zvb

Note that the subgroup is isomorphic to Zb.

Definition 15.2.2. An orbit of a point x0 ∈ Rb under the actionof a lattice L is the subset of Rb given by the collection of elements

{x0 + g : g ∈ L} .These can also be viewed as the cosets of the lattice in Rb.

Definition 15.2.3. The quotient

Rb/L

is called a b-torus.

At this point tori are understood at the group-theoretic level as inthe case of the circle R/L.

15.3. Fundamental domain

Definition 15.3.1. A fundamental domain for the torus Rb/L is aclosed set F ⊆ Rb satisfying the following three conditions:

• every orbit meets F in at least one point;• every orbit meets the interior Int(F ) of F in at most one point;• the boundary ∂F is of zero b-dimensional volume (and can bethought of as a union of (b− 1)-dimensional hyperplanes).

In the literature, one often replaces “n-dimensional volume” by n-dimensional “Lebesgue measure”.

Example 15.3.2. The interval [0, α] is a fundamental domain forthe circle R/Lα.

Example 15.3.3. The parallelepiped spanned by a collection ofbasis vectors for L ⊆ Rb is a fundamental domain fot L.

More concretely, consider the following example.

15.4. LATTICES IN THE PLANE 153

Example 15.3.4. The vectors e1 and e2 in R2 span the unit squarewhich is a fundamental domain for the lattice Z2 ⊆ R2 = C of Gaussianintegers.

Example 15.3.5. Consider the vectors v = (1, 0) and w = (12,√32)

in R2 = C. Their span is a parallelogram giving a fundamental domainfor the lattice of Eisenstein integers (see Example 15.4.2).

Definition 15.3.6. The total volume of the b-torus Rb/L is bydefinition the b-volume of a fundamental domain.

It is shown in advanced calculus that the total volume thus definedis independent of the choice of a fundamental domain.

15.4. Lattices in the plane

Let b = 2. Every lattice L ⊆ R2 is of the form

L = SpanZ(v, w) ⊆ R2,

where {v, w} is a linearly independent set. For example, let α and βbe nonzero reals. Set

Lα,β = SpanZ(αe1, βe2) ⊆ R2.

This lattice admits an orthogonal basis, namely {αe1, βe2}.

Example 15.4.1 (Gaussian integers). For the standard lattice Zb ⊆Rb, the torus Tb = Rb/Zb satisfies vol(Tb) = 1 as it has the unit cubeas a fundamental domain.

In dimension 2, the resulting lattice in C = R2 is called the Gaussianintegers LG. It contains 4 elements of least length. These are the fourthroots of unity. We have

LG = SpanZ(1, i) ⊆ C.

Example 15.4.2 (Eisenstein integers). Consider the lattice LE ⊆R2 = C spanned by 1 ∈ C and the sixth root of unity e

2πi6 ∈ C:

LE = SpanZ(eiπ/3, 1) = Z eiπ/3 + Z 1 ⊆ C. (15.4.1)

The resulting lattice is called the Eisenstein integers. The torus T2 =

R2/LE satisfies area(T2) =√32. The Eisenstein lattice contains 6 ele-

ments of least length, namely all the sixth roots of unity.


15.5. Successive minima of a lattice

Let B be Euclidean space, and let ‖ ‖ be the Euclidean norm.Let L ⊆ (B, ‖ ‖) be a lattice, i.e., span of a collection of b linearlyindependent vectors where b = dim(B).

Definition 15.5.1. The first successive minimum, λ1(L, ‖ ‖) is theleast length of a nonzero vector in L.

We can express the definition symbolically by means of the formula

λ1(L, ‖ ‖) = min{‖v1‖

∣∣ v1 ∈ L \ {0}}.

We illustrate the geometric meaning of λ1 in terms of the circle ofTheorem 15.1.1.

Theorem 15.5.2. Consider a lattice L ⊆ R. Then the circle R/Lsatisfies

length(R/L) = λ1(L).

Proof. This follows by choosing the fundamental domain F =[0, α] where α = λ1(L), so that L = αZ, cf. Example 15.2 above. �

Remark 15.5.3. When α = 1, we can choose a representative fromthe orbit of x to be the fractional part {x} of x.

Definition 15.5.4. For k = 2, define the second successive min-imum of the lattice L with rank(L) ≥ 2 as follows. Given a pair ofvectors S = {v, w} in L, define the size2 |S| of S by setting

|S| = max(‖v‖, ‖w‖).Then the second successive minimum, λ2(L, ‖ ‖) is the least size of apair of non-proportional vectors in L:

λ2(L) = infS|S|,

where S runs over all linearly independent (i.e. non-proportional) pairsof vectors {v, w} ⊆ L.

Example 15.5.5. For both the Gaussian and the Eisenstein integerswe have λ1 = λ2 = 1.

Example 15.5.6. For the lattice Lα,β we have λ1(Lα,β) = min(|α|, |β|)and λ2(Lα,β) = max(|α|, |β|).

2Quotation marks: merka’ot.

15.7. SPHERE AND TORUS AS TOPOLOGICAL SURFACES 155

15.6. Gram matrix

The volume of the torus Rb/L (see Definition 15.2.3) is also calledthe covolume of the lattice L. It is by definition the volume of a funda-mental domain for L, e.g. a parallelepiped spanned by a Z-basis for L.

The Gram matrix was defined in Definition 5.7.2.

Theorem 15.6.1. Let L ⊂ Rb be a lattice spanned by linearly inde-pendent vectors (v1, . . . , vb). Then the volume of the torus Rb/L is thesquare root of the determinant of the Gram matrix Gram(v1, . . . , vb).

In geometric terms, the parallelepiped P spanned by the vectors {vi}satisfies

vol(P ) =√

det(Gram(S)). (15.6.1)

Proof. Let A be the square matrix whose columns are the columnvectors v1, v2 . . . , vn in Rn. It is shown in linear algebra that

vol(P ) = |det(A)|.Let B = AtA, and let B = (bij). Then

bij = vti vj = 〈vi, vj〉Hence B = Gram(S). Thus

det(Gram(S)) = det(At A) = det(A)2 = vol(P )2


15.7. Sphere and torus as topological surfaces

The topology of surfaces will be discussed in more detail in Chap-ter 18. For now, we will recall that a compact surface can be eitherorientable or non-orientable. An orientable surface is characterizedtopologically by its genus, i.e. number of “handles”.

Recall that the unit sphere in R3 can be represented implicitly bythe equation

x2 + y2 + z2 = 1.

Parametric representations of surfaces are discussed in Section 5.4.

Example 15.7.1. The sphere has genus 0 (no handles).

Theorem 15.7.2. The 2-torus is characterized topologically in oneof the following four equivalent ways:

(1) the Cartesian product of a pair of circles: S1 × S1;(2) the surface of revolution in R3 obtained by starting with the

following circle in the (x, z)-plane: (x − 10)2 + z2 = 1 (forexample), and rotating it around the z-axis;


(3) a quotient R2/L of the plane by a lattice L;(4) a compact 2-dimensional manifold of genus 1.

The equivalence between items (2) and (3) can be seen by markinga pair of generators of L by different color, and using the same colors toindicate the corresponding circles on the embedded torus of revolution,as follows:

Figure 15.7.1. Torus viewed by means of its lattice(left) and by means of a Euclidean embedding (right)

Note by comparison that a circle can be represented either by itsfundamental domain which is [0, 2π] (with endpoints identified), or asthe unit circle embedded in the plane.3

15.8. Standard fundamental domain

We will discuss the case b = 2 in detail. An important role is playedin this dimension by the standard fundamental domain.

3The Hermite constant γb is defined in one of the following two equivalent ways:

(1) γb is the square of the maximal first successive minimum λ1, among alllattices of unit covolume;

(2) γb is defined by the formula

√γb = sup

{λ1(L)

vol(Rb/L)1b

∣∣∣∣L ⊆ (Rb, ‖ ‖)}, (15.7.1)

where the supremum is extended over all lattices L in Rb with a Euclideannorm ‖ ‖.

A lattice realizing the supremum may be thought of as the one realizing the densestpacking in Rb when we place the balls of radius 1

2λ1(L) at the points of L. Indimensions b ≥ 3, the Hermite constants are harder to compute, but explicit values(as well as the associated critical lattices) are known for small dimensions, e.g. γ3 =

213 = 1.2599..., while γ4 =

√2 = 1.4142....

15.9. CONFORMAL PARAMETER τ OF A LATTICE 157

Definition 15.8.1. The standard fundamental domain, denoted D,is the set

D =

{z ∈ C

∣∣∣∣ |z| ≥ 1, |Re(z)| ≤ 1

2, Im(z) > 0

}(15.8.1)

cf. [Ser73, p. 78].

The domain D a fundamental domain for the action of PSL(2,Z)in the upperhalf plane of C.

Lemma 15.8.2. Multiplying a lattice L ⊆ C by nonzero complexnumbers does not change the value of the quotient

λ1(L)2

area(C/L).

Proof. We write such a complex number as reiθ. Note that multi-plication by reiθ can be thought of as a composition of a scaling by thereal factor r, and rotation by angle θ. The rotation is an isometry (con-gruence) that preserves all lengths, and in particular the length λ1(L)and the area of the quotient torus.

Meanwhile, multiplication by r results in a cancellation

λ1(rL)2

area(C/rL)=

(rλ1(L))2

r2 area(C/L)=

r2λ1(L)2

r2 area(C/L)=

λ1(L)2

area(C/L),


15.9. Conformal parameter τ of a lattice

Two lattices in C are said to be similar if one is obtained from theother by multiplication by a nonzero complex number.

Theorem 15.9.1. Every lattice in C is similar to a lattice spannedby {τ, 1} where τ is in the standard fundamental domain D of (15.8.1).The value τ = eiπ/3 corresponds to the Eisenstein integers (15.4.1).

Proof. Let L ⊆ C be a lattice. Choose a “shortest” vector z ∈L, i.e. we have |z| = λ1(L). By Lemma 15.8.2, we may replace thelattice L by the lattice z−1L.

Thus, we may assume without loss of generality that the com-plex number +1 ∈ C is a shortest element in the lattice L. Thuswe have λ1(L) = 1. Now complete the element +1 to a Z-basis

{τ ,+1}for L. Here we may assume, by replacing τ by−τ if necessary, that Im(τ) >0.


Now consider the real part Re(τ). We adjust the basis by adding asuitable integer k to τ :

τ = τ − k where k =[Re(τ) + 1

2

](15.9.1)

(the brackets denote the integer part), so it satisfies the condition

−1

2≤ Re(τ) ≤ 1

2.

Since τ ∈ L, we have |τ | ≥ λ1(L) = 1. Therefore the element τ lies inthe standard fundamental domain (15.8.1). �

Example 15.9.2. For the “rectangular” lattice Lα,β = SpanZ(α, βi),we obtain

τ(Lα,β) =

{ |β||α|i if |β| > |α||α||β| i if |α| > |β|.

Corollary 15.9.3. Let b = 2. Then we have the following valuefor the Hermite constant: γ2 = 2√

3= 1.1547.... The corresponding

optimal lattice is homothetic to the Z-span of cube roots of unity in C(i.e. the Eisenstein integers).

Proof. Choose τ as in (15.9.1) above. The pair

{τ,+1}

is a basis for the lattice. The imaginary part satisfies Im(τ) ≥√32, with

equality possible precisely for

τ = eiπ3 or τ = ei

2π3 .

Moreover, if τ = r exp(iθ), then

|τ | sin θ = Im(τ) ≥√3

2.

The proof is concluded by calculating the area of the parallelogramin C spanned by τ and +1;

λ1(L)2

area(C/L)=

1

|τ | sin θ ≤ 2√3,


Definition 15.9.4. A τ ∈ D is said to be the conformal parameterof a flat torus T 2 if T 2 is similar to a torus C/L where L = Zτ + Z1.

15.10. CONFORMAL PARAMETER τ OF TORI OF REVOLUTION 159

15.10. Conformal parameter τ of tori of revolution

The results of Section 13.6 have the following immediate conse-quence.

Corollary 15.10.1. Consider a torus of revolution in R3 formedby rotating a Jordan curve of length L > 0, with unit speed param-etisation (f(φ), g(φ)) where φ ∈ [0, L]. Then the torus is conformallyequivalent to a flat torus

R2/Lc,d.

Here R2 is the (θ, ψ)-plane, where ψ is the antiderivative of 1f(φ)

as in

Section 13.6; while the rectangular lattice Lc,d ⊂ R2 is spanned by theorthogonal vectors c ∂

∂θand d ∂

∂ψ, so that

Lc,d = Span

(c∂

∂θ, d

∂

∂ψ

)= cZ⊕ dZ,

where c = 2π and d =∫ L0

dφf(φ)

.

Figure 15.10.1. Torus: lattice (left) and embedding (right)

In Section 15.8 we showed that every flat torus C/L is similar tothe torus spanned by τ ∈ C and 1 ∈ C, where τ is in the standardfundamental domain

D = {z = x+ iy ∈ C : |x| ≤ 12, y > 0, |z| ≥ 1}.

Definition 15.10.2. The parameter τ is called the conformal pa-rameter of the torus.

Corollary 15.10.3. The conformal parameter τ of a torus of rev-olution is pure imaginary:

τ = iσ2


of absolute value

σ2 = max

{c

d,d

c

}≥ 1.

Proof. The proof is immediate from the fact that the lattice isrectangular. �

15.11. θ-loops and φ-loops on tori of revolution

Consider a torus of revolution (T 2, g) generated by a Jordan curve Cin the (x, z)-plane, i.e., by a simple loop C, parametrized by a pair offunctions f(φ), g(φ), so that x = f(φ) and z = g(φ).

Definition 15.11.1. A φ-loop on the torus is a simple loop ob-tained by fixing the coordinate θ (i.e., the variable φ is changing). A θ-loop on the torus is a simple loop obtained by fixing the coordinate φ(i.e., the variable θ is changing).

Proposition 15.11.2. All φ-loops on the torus of revolution havethe same length equal to the length L of the generating curve C (seeCorollary 15.10.1).

Proof. The surface is rotationally invariant. In other words, allrotations around the z-axis are isometries. Therefore all φ-loops havethe same length. �

Proposition 15.11.3. The θ-loops on the torus of revolution havevariable length, depending on the φ-coordinate of the loop. Namely, thelength is 2πx = 2πf(φ).

Proof. The proof is immediate from the fact that the function f(φ)gives the distance r to the z-axis. �

Definition 15.11.4. We denote by λφ the (common) length of allφ-loops on a torus of revolution.

Definition 15.11.5. We denote by λθminthe least length of a θ-

loop on a torus of revolution, and by λθmaxthe maximal length of such

a θ-loop.

15.12. Tori generated by round circles

Let a, b > 0. We assume a > b so as to obtain tori that are em-bedded in 3-space. We consider the 2-parameter family ga,b of toriof revolution in 3-space with circular generating loop. The torus ofrevolution ga,b generated by a round circle is the locus of the equation

(r − a)2 + z2 = b2, (15.12.1)

15.13. CONFORMAL PARAMETER OF TORI OF REVOLUTION, RESIDUES161

where r =√x2 + y2. Note that the angle θ of the cylindrical coordi-

nates (r, θ, z) does not appear in the equation (15.12.1). The torus isobtained by rotating the circle

(x− a)2 + z2 = b2 (15.12.2)

around the z-axis in R3. The torus admits a parametrisation in termsof the functions4 f(φ) = a + b cosφ and g(φ) = b sinφ. Namely, wehave

x(θ, φ) = ((a+ b cos φ) cos θ, (a+ b cosφ) sin θ, b sinφ). (15.12.3)

Here the θ-loop (see Section 15.11) has length 2π(a+ b cos φ). Theshortest θ-loop is therefore of length

λθmin= 2π(a− b),

and the longest one is

λθmax= 2π(a+ b).

Meanwhile, the φ-loop has length

λφ = 2πb.

15.13. Conformal parameter of tori of revolution, residues

This section is optional.We would like to compute the conformal parameter τ of the stan-

dard tori as in (15.12.3). We first modify the parametrisation so as toobtain a generating curve parametrized by arclength:

f(ϕ) = a+ b cosϕ

b, g(ϕ) = b sin

ϕ

b, (15.13.1)

where ϕ ∈ [0, L] with L = 2πb.

Theorem 15.13.1. The corresponding flat torus is given by the

lattice L in the (θ, ψ) plane of the form L = SpanZ

(c ∂∂θ, d ∂

∂φ

)

where c = 2π and d = 2π√(a/b)2−1

. Thus the conformal parameter τ

of the flat torus satisfies τ = imax(((a/b)2 − 1)−1/2, ((a/b)2 − 1)1/2

).

Proof. By Corollary 15.10.3, replacing ϕ by ϕ(ψ) produces isother-mal coordinates (θ, ψ) for the torus generated by (15.13.1), where ψ =∫

dϕf(ϕ)

=∫

dϕa+b cos ϕ

b, and therefore the flat metric is defined by a lattice

in the (θ, ψ) plane with c = 2π and d =∫ L=2πb

0dϕ

a+b cos ϕb. Changing

the variable to to φ = ϕbwe obtain d =

∫ 2π

0dφ

(a/b)+cos φ, where a/b > 1.

4We use φ here and ϕ for the modified arclength parameter in the next section


Let α = a/b. Now the integral is the real part Re of the complex

integral d =∫ 2π

0dφ

α+cosφ=∫

dφα+Re(eiφ)

. Thus

d =

∫2dφ

2α+ eiφ + e−iφ. (15.13.2)

The change of variables z = eiφ yields dφ = −idzz

and along the circle

we have d =∮ −2idz

z(2α+z+z−1)=∮ −2idz

z2+2αz+1=∮ −2idz

(z−λ1)(z−λ2) , where λ1 =

−α+√α2 − 1 and λ2 = −α−

√α2 − 1. The root λ2 is outside the unit

circle. Hence we need the residue at λ1 to apply the residue theorem.The residue at λ1 equals Resλ1 = −2i

λ1−λ2 = −2i2√α2−1

= −i√α2−1

. The inte-

gral is determined by the residue theorem in terms of the residue at thepole z = λ1. Therefore the lattice parameter d from Corollary 15.10.3can be computed from (15.13.2) as d =

(2πiResλ1

)= 2π√

(α)2−1. proving

the theorem. �

CHAPTER 16

A hyperreal view

16.1. Successive extensions N, Z, Q, R, ∗R

Our reference for true infinitesimal calculus is Keisler’s textbook[Ke74], downloadable athttp://www.math.wisc.edu/~keisler/calc.html

We start by motivating the familiar sequence of extensions of num-ber systems

N → Z → Q → Rin terms of their applications in arithmetic, algebra, and geometry.Each successive extension is introduced for the purpose of solving prob-lems, rather than enlarging the number system for its own sake. Thus,the extension Q ⊆ R enables one to express the length of the diagonalof the unit square and the area of the unit disc in our number system.

The familiar continuum R is an Archimedean continuum, in thesense that it satisfies the following Archimedean property.

Definition 16.1.1. An ordered field extending N is said to satisfythe Archimedean property if

(∀ǫ > 0)(∃n ∈ N) [nǫ > 1] .

We will provisionally denote the real continuum A where “A” standsfor Archimedean. Thus we obtain a chain of extensions

N → Z → Q → A,

as above. In each case one needs an enhanced ordered1 number systemto solve an ever broader range of problems from algebra or geometry.

The next stage is the extension

N → Z → Q → A → B,

where B is a Bernoullian continuum containing infinitesimals, definedas follows.

Definition 16.1.2. A Bernoullian extension of R is any properextension which is an ordered field.

1sadur

163

http://www.math.wisc.edu/~keisler/calc.html

164 16. A HYPERREAL VIEW

Any Bernoullian extension allows us to define infinitesimals and dointeresting things with those. But things become really interesting ifwe assume the Transfer Principle (Section 16.4), and work in a truehyperreal field, defined as in Definition 16.3.4 below. We will providesome motivating comments for the transfer principle in Section 16.3.

16.2. Motivating discussion for infinitesimals

Infinitesimals can be motivated from three different angles: geo-metric, algebraic, and arithmetic/analytic.

θ

Figure 16.2.1. Horn angle θ is smaller than every rec-tilinear angle

(1) Geometric (horn angles): Some students have expressedthe sentiment that they did not understand infinitesimals untilthey heard a geometric explanation of them in terms of whatwas classically known as horn angles. A horn angle is thecrivice between a circle and its tangent line at the point oftangency. If one thinks of this crevice as a quantity, it is easy toconvince oneself that it should be smaller than every rectilinearangle (see Figure 16.2.1). This is because a sufficiently smallarc of the circle will be contained in the convex region cutout by the rectilinear angle no matter how small. When onerenders this in terms of analysis and arithmetic, one gets apositive quantity smaller than every positive real number. Wecite this example merely as intuitive motivation (our actualconstruction is different).

16.3. INTRODUCTION TO THE TRANSFER PRINCIPLE 165

(2) Algebraic (passage from ring to field): The idea is torepresent an infinitesimal by a sequence tending to zero. Onecan get something in this direction without reliance on anyform of the axiom of choice. Namely, take the ring S of allsequences of real numbers, with arithmetic operations definedterm-by-term. Now quotient the ring S by the equivalencerelation that declares two sequences to be equivalent if theydiffer only on a finite set of indices. The resulting object S/Kis a proper ring extension of R, where R is embedded by meansof the constant sequences. However, this object is not a field.For example, it has zero divisors. But quotienting it furtherin such a way as to get a field, by extending the kernel K to amaximal ideal K ′, produces a field S/K ′, namely a hyperrealfield.

(3) Analytic/arithmetic: One can mimick the construction ofthe reals out of the rationals as the set of equivalence classesof Cauchy sequences, and construct the hyperreals as equiva-lence classes of sequences of real numbers under an appropriateequivalence relation.

16.3. Introduction to the transfer principle

The transfer principle is a type of theorem that, depending on thecontext, asserts that rules, laws or procedures valid for a certain num-ber system, still apply (i.e., are “transfered”) to an extended numbersystem.

Example 16.3.1. The familiar extension Q ⊆ R preserves the prop-erty of being an ordered field.

Example 16.3.2. To give a negative example, the extension R ⊆R∪{±∞} of the real numbers to the so-called extended reals does notpreserve the property of being an ordered field.

The hyperreal extension R ⊆ ∗R (defined below) preserves all first-order properties (i.e., properties involving quantification over elementsbut not over sets.

Example 16.3.3. The formula sin2 x+cos2 x = 1, true over R for allreal x, remains valid over ∗R for all hyperreal x, including infinitesimaland infinite values of x ∈ ∗R.

Thus the transfer principle for the extension R ⊆ ∗R is a theo-rem asserting that any statement true over R is similarly true over ∗R,


and vice versa. Historically, the transfer principle has its roots in theprocedures involving Leibniz’s Law of continuity.2

We will explain the transfer principle in several stages of increasingdegree of abstraction. More details can be found in Section 16.4.

Definition 16.3.4. An ordered field B, properly including the fieldA = R of real numbers (so that A $ B) and satisfying the TransferPrinciple, is called a hyperreal field. If such an extended field B is fixedthen elements of B are called hyperreal numbers,3 while the extendedfield itself is usually denoted ∗R.

Theorem 16.3.5. Hyperreal fields exist.

For example, a hyperreal field can be constructed as the quotient ofthe ring RN of sequences of real numbers, by an appropriate maximalideal.

Definition 16.3.6. A positive infinitesimal is a positive hyperrealnumber ǫ such that

(∀n ∈ N) [nǫ < 1]

More generally, we have the following.

Definition 16.3.7. A hyperreal number ε is said to be infinitelysmall or infinitesimal if

−a < ε < a

for every positive real number a.

In particular, one has ε < 12, ε < 1

3, ε < 1

4, ε < 1

5, etc. If ε > 0

is infinitesimal then N = 1εis positive infinite, i.e., greater than every

real number.A hyperreal number that is not an infinite number are called finite.

Sometimes the term limited is used in place of finite.Keisler’s textbook exploits the technique of representing the hyper-

real line graphically by means of dots indicating the separation betweenthe finite realm and the infinite realm. One can view infinitesimals withmicroscopes as in Figure 16.6.1. One can also view infinite numberswith telescopes as in Figure 16.3.1. We have an important subset

{finite hyperreals} ⊆ ∗R

2Leibniz’s theoretical strategy in dealing with infinitesimals was analyzed ina number of detailed studies recently, which found Leibniz’ strategy to be morerobust than George Berkeley’s flawed critique thereof.

3Similar terminology is used with regard to integers and hyperintegers.

16.3. INTRODUCTION TO THE TRANSFER PRINCIPLE 167

Finite

−2 −1 0 1 2

Positive

infinite

Infinite

telescope1ǫ− 1

1/ǫ

1ǫ+ 1

Figure 16.3.1. Keisler’s telescope

which is the domain of the function called the standard part function(also known as the shadow) which rounds off each finite hyperreal tothe nearest real number (for details see Section 16.6).

Example 16.3.8. Slope calculation of y = x2 at x0. Here we usestandard part function (also called the shadow)

st : {finite hyperreals} → R

(for details concerning the shadow see Section 16.6). If a curve is de-fined by y = x2 we wish to find the slope at the point x0. To this endwe use an infinitesimal x-increment ∆x and compute the correspond-ing y-increment

∆y = (x0+∆x)2−x20 = (x0+∆x+x0)(x+∆x−x0) = (2x0+∆x)∆x.

The corresponding “average” slope is therefore

∆y

∆x= 2x0 +∆x

which is infinitely close to 2x, and we are naturally led to the definitionof the slope at x0 as the shadow of ∆y

∆x, namely st

(∆y∆x

)= 2x0.


The extension principle expresses the idea that all real objects havenatural hyperreal counterparts. We will be mainly interested in sets,functions and relations. We then have the following extension principle.

Extension principle. The order relation on the hyperreals con-tains the order relation on the reals. There exists a hyperreal numbergreater than zero but smaller than every positive real. Every set D ⊆ Rhas a natural extension ∗D ⊆ ∗R. Every real function f with domain Dhas a natural hyperreal extension ∗f with domain ∗D.4

Here the naturality of the extension alludes to the fact that suchan extension is unique, and the coherence refers to the fact that thedomain of the natural extension of a function is the natural extensionof its domain.

A positive infinitesimal is a positive hyperreal smaller than everypositive real. A negative infinitesimal is a negative hyperreal greaterthan every negative real. An arbitrary infinitesimal is either a positiveinfinitesimal, a negative infinitesimal, or zero.

Ultimately it turns out counterproductive to employ asterisks forhyperreal functions (in fact we already dropped it in equation (16.4.1)).

16.4. Transfer principle

Definition 16.4.1. The Transfer Principle asserts that every first-order statement true over R is similarly true over ∗R, and vice versa.

Here the adjective first-order alludes to the limitation on quantifi-cation to elements as opposed to sets.

Listed below are a few examples of first-order statements.

Example 16.4.2. The commutativity rule for addition x+y = y+xis valid for all hyperreal x, y by the transfer principle.

Example 16.4.3. The formula

sin2 x+ cos2 x = 1 (16.4.1)

is valid for all hyperreal x by the transfer principle.

Example 16.4.4. The statement

0 < x < y =⇒ 0 <1

y<

1

x(16.4.2)

holds for all hyperreal x, y.

4Here the noun principle (in extension principle) means that we are going toassume that there is a function ∗f : B → B which satisfies certain properties.It is a separate problem to define B which admits a coherent definition of ∗f forall f : A → A, to be solved below.

16.5. ORDERS OF MAGNITUDE 169

Example 16.4.5. The characteristic function χQ of the rationalnumbers equals 1 on rational inputs and 0 on irrational inputs. Bythe transfer principle, its natural extension ∗χQ = χ∗Q will be 1 onhyperrational numbers ∗Q and 0 on hyperirrational numbers (namely,numbers in the complement ∗R \ ∗Q).

To give additional examples of real statements to which transferapplies, note that all ordered field -statements are subject to Trans-fer. As we will see below, it is possible to extend Transfer to a muchbroader category of statements, such as those containing the functionsymbols exp or sin or those that involve infinite sequences of reals.

A hyperreal number x is finite if there exists a real number r suchthat |x| < r. A hyperreal number is called positive infinite if it isgreater than every real number, and negative infinite if it is smallerthan every real number.

16.5. Orders of magnitude

Hyperreal numbers come in three orders of magnitude: infinitesi-mal, appreciable, and infinite. A number is appreciable if it is finitebut not infinitesimal. In this section we will outline the rules for ma-nipulating hyperreal numbers.

To give a typical proof, consider the rule that if ǫ is positive in-finitesimal then 1

ǫis positive infinite. Indeed, for every positive real r

we have 0 < ǫ < r. It follows from (16.4.2) by transfer that that 1ǫis

greater than every positive real, i.e., that 1ǫis infinite.

Let ǫ, δ denote arbitrary infinitesimals. Let b, c denote arbitraryappreciable numbers. Let H,K denote arbitrary infinite numbers. Wehave the following theorem.

Theorem 16.5.1. We have the following rules for addition:

• ǫ+ δ is infinitesimal;• b+ ǫ is appreciable;• b+ c is finite (possibly infinitesimal);• H + ǫ and H + b are infinite.

We have the following rules for products.

• ǫδ and bǫ are infinitesimal;• bc is appreciable;• Hb and HK are infinite.

We have the following rules for quotients.

• ǫb, ǫH, bH

are infinitesimal;

• bcis appreciable;


• bǫ, Hǫ, Hbare infinite provided ǫ 6= 0.

We have the following rules for roots, where n is a standard naturalnumber.

• if ǫ > 0 then n√ǫ is infinitesimal;

• if b > 0 then n√b is appreciable;

• if H > 0 then n√H is infinite.

Note that the traditional topic of the so-called “indeterminate forms”can be treated without introducing any ad-hoc terminology by meansof the following remark.

Remark 16.5.2. We have no rules in certain cases, such as ǫδ, HK,Hǫ,

and H +K.

These cases correspond to what are known since Moigno5 as inde-terminate forms.

Theorem 16.5.3. Arithmetic operations on the hyperreal numbersare governed by the following rules.

(1) every hyperreal number between two infinitesimals is infinites-imal.

(2) Every hyperreal number which is between two finite hyperrealnumbers, is finite.

(3) Every hyperreal number which is greater than some positiveinfinite number, is positive infinite.

(4) Every hyperreal number which is less than some negative infi-nite number, is negative infinite.

Example 16.5.4. The difference√H + 1 −

√H − 1 (where H is

infinite) is infinitesimal. Namely,

√H + 1−

√H − 1 =

(√H + 1−

√H − 1

) (√H + 1 +

√H − 1

)(√

H + 1 +√H − 1

)

=H + 1− (H − 1)(√H + 1 +

√H − 1

)

=2√

H + 1 +√H − 1

is infinitesimal. Once we introduce limits, this example can be refor-mulated as follows: limn→∞

(√n + 1−

√n− 1

)= 0.

5Elaborate on this historical note.

16.7. DIFFERENTIATION 171

Definition 16.5.5. Two hyperreal numbers a, b are said to be in-finitely close, written

a ≈ b,

if their difference a− b is infinitesimal.

It is convenient also to introduce the following terminology andnotation.

Definition 16.5.6. Two nonzero hyperreal numbers a, b are saidto be adequal, written

a pq b,

if either ab≈ 1 or a = b = 0.

Note that the relation sin x ≈ x for infinitesimal x is immediatefrom the continuity of sine at the origin (in fact both sides are infin-itely close to 0), whereas the relation sin x pq x is a subtler relationequivalent to the computation of the first order Taylor approximationof sine.

16.6. Standard part principle

Theorem 16.6.1 (Standard Part Principle). Every finite hyperrealnumber x is infinitely close to an appropriate real number.

Proof. The result is generally true for an arbitrary proper orderedfield extension E of R. Indeed, if x ∈ E is finite, then x induces aDedekind cut on the subfield Q ⊆ R ⊆ E via the total order of E. Thereal number corresponding to the Dedekind cut is then infinitely closeto x. �

The real number infinitely close to x is called the standard part, orshadow, denoted st(x), of x.

We will use the notation ∆x, ∆y for infinitesimals.

Remark 16.6.2. There are three consecutive stages in a typicalcalculation: (1) calculations with hyperreal numbers, (2) calculationwith standard part, (3) calculation with real numbers.

16.7. Differentiation

An infinitesimal increment ∆x can be visualized graphically bymeans of a microscope as in the Figure 16.7.1.

The slope s of a function f at a real point a is defined by setting

s = st

(f(a+∆x)− f(a)

∆x

)


r r + β r + γr + α

st

−1

−1

0

0

1

1

2

2

3

3

4

4r

Figure 16.6.1. The standard part function, st, “roundsoff” a finite hyperreal to the nearest real number. The func-tion st is here represented by a vertical projection. Keisler’s“infinitesimal microscope” is used to view an infinitesimalneighborhood of a standard real number r, where α, β,and γ represent typical infinitesimals. Courtesy ofWikipedia.

whenever the shadow exists (i.e., the ratio is finite) and is the samefor each nonzero infinitesimal ∆x. The construction is illustrated inFigure 16.7.2.

Definition 16.7.1. Let f be a real function of one real variable.The derivative of f is the new function f ′ whose value at a real x isthe slope of f at x. In symbols,

f ′(x) = st

(f(x+∆x)− f(x)

∆x

)

whenever the slope exists.

Equivalently, we can write f ′(x) ≈ f(x+∆x)−f(x)∆x

. When y = f(x)we define a new dependent variable ∆y by settting

∆y = f(x+∆x)− f(x)

called the y-increment, so we can write the derivative as st(∆y∆x

).

16.7. DIFFERENTIATION 173

a

a a+∆x

Figure 16.7.1. Infinitesimal increment ∆x under the microscope

Example 16.7.2. If f(x) = x2 we obtain the derivative of y = f(x)by the following direct calculation:

f ′(x) ≈ ∆y

∆x

=(x+∆x)2 − x2

∆x

=(x+∆x− x)(x+∆x+ x)

∆x

=∆x(2x+∆x)

∆x= 2x+∆x

≈ 2x.


a

|f(a+∆x)− f(a)|

a a+∆x

Figure 16.7.2. Defining slope of f at a

Given a function y = f(x) one defines the dependent variable ∆y =f(x+∆x)−f(x) as above. One also defines a new dependent variable dyby setting dy = f ′(x)∆x at a point where f is differentiable, and setsfor symmetry dx = ∆x. Note that we have

dy pq ∆y

as in Definition 16.5.6 whenever dy 6= 0. We then have Leibniz’s nota-tion

f ′(x) =dy

dxand rules like the chain rule acquire an appealing form.

CHAPTER 17

Global and systolic geometry

17.1. Definition of systole

The unit circle S1 ⊂ C bounds the unit disk D. A loop on asurface M is a continuous map S1 →M .

Definition 17.1.1. A loop S1 → M is called contractible if themap f extends from S1 to the disk D by means of a continuous map F :D →M .

Thus the restriction of F to S1 is f .The notions of contractible loops and simply connected spaces were

reviewed in more detail in Section 18.1.

Definition 17.1.2. A loop is called noncontractible if it is notcontractible.

Definition 17.1.3. Given a metric g onM , we will denote by sys1(g),the infimum of lengths, referred to as the “systole” of g, of a non-contractible loop β in a compact, non-simply-connected Riemannianmanifold (M, g):

sys1(g) = infβlength(β), (17.1.1)

where the infimum is over all noncontractible loops β in M . In graphtheory, a similar invariant is known as the girth [Tu47].1

It can be shown that for a compact Riemannian manifold, the in-fimum is always attained, cf. Theorem 17.18.1. A loop realizing theminimum is necessarily a simple closed geodesic.

In systolic questions about surfaces, integral-geometric identitiesplay a particularly important role. Roughly speaking, there is an inte-gral identity relating area on the one hand, and an average of energiesof a suitable family of loops, on the other. By the Cauchy-Schwarzinequality, there is an inequality relating energy and length squared,hence one obtains an inequality between area and the square of thesystole.

1The notion of systole expressed by (17.1.1) is unrelated to the systolic arraysof [Ku78].

175

176 17. GLOBAL AND SYSTOLIC GEOMETRY

Such an approach works both for the Loewner inequality (17.2.1)and Pu’s inequality (17.1.6) (biographical notes on C. Loewner andP. Pu appear in respectively). One can prove an inequality for theMobius band this way, as well [Bl61b].

Here we prove the two classical results of systolic geometry, namelyLoewner’s torus inequality as well as Pu’s inequality for the real pro-jective plane.

17.1.1. Three systolic invariants. The material in this subsec-tion is optional.

LetM be a Riemannian manifold. We define the homology 1-systole

sys1(M) (17.1.2)

by minimizing vol(α) over all nonzero homology classes. Namely, sys1(M)is the least length of a loop C representing a nontrivial homologyclass [C] in H1(M ;Z).

We also define the stable homology systole

stsys1(M) = λ1 (H1(M)/T1, ‖ ‖) , (17.1.3)

namely by minimizing the stable norm ‖ ‖ of a class of infinite order(see Definition 18.13.1 for details).

Remark 17.1.4. For the real projective plane, these two systolic in-variants are not the same. Namely, the homology systole sys1 equals theleast length of a noncontractible loop (which is also nontrivial homolog-ically), while the stable systole is infinite being defined by a minimumover an empty set.

Recall the following example from the previous section:

Example 17.1.5. For an arbitrary metric on the 2-torus T2, the1-systole and the stable 1-systole coincide by Theorem 18.5.3:

sys1(T2) = stsys1(T

2),

for every metric on T2.

Using the notion of a noncontractible loop, we can define the ho-motopy 1-systole

sys1(M) (17.1.4)

as the least length of a non-contractible loop in M .In the case of the torus, the fundamental group Z2 is abelian and

torsionfree, and therefore sys1(T2) = sys1(T

2), so that all three invari-ants coincide in this case.

17.1. DEFINITION OF SYSTOLE 177

17.1.2. Isoperimetric inequality and Pu’s inequality. Thematerial in this section is optional.

Pu’s inequality can be thought of as an “opposite” isoperimetricinequality, in the following precise sense.

The classical isoperimetric inequality in the plane is a relation be-tween two metric invariants: length L of a simple closed curve in theplane, and area A of the region bounded by the curve. Namely, everysimple closed curve in the plane satisfies the inequality

A

π≤(L

2π

)2

.

This classical isoperimetric inequality is sharp, insofar as equality isattained only by a round circle.

In the 1950’s, Charles Loewner’s student P. M. Pu [Pu52] provedthe following theorem. Let RP2 be the real projective plane endowedwith an arbitrary metric, i.e. an embedding in some Rn. Then

(L

π

)2

≤ A

2π, (17.1.5)

where A is its total area and L is the length of its shortest non-contractible loop. This isosystolic inequality, or simply systolic inequal-ity for short, is also sharp, to the extent that equality is attained onlyfor a metric of constant Gaussian curvature, namely antipodal quotientof a round sphere, cf. Section 17.19. In our systolic notation (17.1.1),Pu’s inequality takes the following form:

sys1(g)2 ≤ π

2area(g), (17.1.6)

for every metric g on RP2. See Theorem 17.21.2 for a discussion of theconstant. The inequality is proved in Section 20.5. Pu’s inequality canbe generalized as follows. We will say that a surface is aspherical if itis not a 2-sphere.

Theorem 17.1.6. Every aspherical surface (M, g) satisfies the op-timal bound (17.1.6), attained precisely when, on the one hand, thesurface M is a real projective plane, and on the other, the metric g isof constant Gaussian curvature.

The extension to aspherical surfaces follows from Gromov’s inequal-ity (17.1.7) below (by comparing the numerical values of the two con-stants). Namely, every aspherical compact surface (M, g) admits ametric ball

B = Bp

(12sys1(g)

)⊆M


of radius 12sys1(g) which satisfies [Gro83, Corollary 5.2.B]

sys1(g)2 ≤ 4

3area(B). (17.1.7)

17.1.3. Hermite and Berge-Martinet constants. The mate-rial in this subsection is optional.

Most of the material in this section has already appeared in earlierchapters.

Let b ∈ N. The Hermite constant γb is defined in one of the followingtwo equivalent ways:

(1) γb is the square of the biggest first successive minimum, cf. Defi-nition 17.22.1, among all lattices of unit covolume;

(2) γb is defined by the formula

√γb = sup

{λ1(L)

vol(Rb/L)1/b

∣∣∣∣L ⊆ (Rb, ‖ ‖)}, (17.1.8)

where the supremum is extended over all lattices L in Rb witha Euclidean norm ‖ ‖.

A lattice realizing the supremum is called a critical lattice. A crit-ical lattice may be thought of as the one realizing the densest packingin Rb when we place balls of radius 1

2λ1(L) at the points of L.

The existence of the Hermite constant, as well as the existence ofcritical lattices, are both nontrivial results [Ca71].

Theorem 15.9.1 provides the value for γ2.

Example 17.1.7. In dimensions b ≥ 3, the Hermite constants areharder to compute, but explicit values (as well as the associated crit-

ical lattices) are known for small dimensions (≤ 8), e.g. γ3 = 213 =

1.2599 . . ., while γ4 =√2 = 1.4142 . . .. Note that γn is asymptotically

linear in n, cf. (17.1.11).

A related constant γ′b is defined as follows, cf. [BeM].

Definition 17.1.8. The Berge-Martinet constant γ′b is defined bysetting

γ′b = sup{λ1(L)λ1(L

∗)∣∣L ⊆ (Rb, ‖ ‖)

}, (17.1.9)

where the supremum is extended over all lattices L in Rb.

Here L∗ is the lattice dual to L. If L is the Z-span of vectors (xi),then L∗ is the Z-span of a dual basis (yj) satisfying 〈xi, yj〉 = δij,cf. relation (17.12.1).

17.2. LOEWNER’S TORUS INEQUALITY 179

Thus, the constant γ′b is bounded above by the Hermite constant γbof (17.1.8). We have γ′1 = 1, while for b ≥ 2 we have the followinginequality:

γ′b ≤ γb ≤2

3b for all b ≥ 2. (17.1.10)

Moreover, one has the following asymptotic estimates:

b

2πe(1 + o(1)) ≤ γ′b ≤

b

πe(1 + o(1)) for b→ ∞, (17.1.11)

cf. [LaLS90, pp. 334, 337]. Note that the lower bound of (17.1.11)for the Hermite constant and the Berge-Martinet constant is noncon-structive, but see [RT90] and [ConS99].

Definition 17.1.9. A lattice L realizing the supremum in (17.1.9)or (17.1.9) is called dual-critical.

Remark 17.1.10. The constants γ′b and the dual-critical latticesin Rb are explicitly known for b ≤ 4, cf. [BeM, Proposition 2.13]. Inparticular, we have γ′1 = 1, γ′2 =

2√3.

Example 17.1.11. In dimension 3, the value of the Berge-Martinet

constant, γ′3 =√

32= 1.2247 . . ., is slightly below the Hermite con-

stant γ3 = 213 = 1.2599 . . .. It is attained by the face-centered cu-

bic lattice, which is not isodual [MilH73, p. 31], [BeM, Proposition2.13(iii)], [CoS94].

This is the end of the three subsections containing optional material.

17.2. Loewner’s torus inequality

Historically, the first lower bound for the volume of a Riemannianmanifold in terms of a systole is due to Charles Loewner. In 1949,Loewner proved the first systolic inequality, in a course on Riemanniangeometry at Syracuse University, cf. [Pu52]. Namely, he showed thefollowing result, whose proof appears in Section 20.2.

Theorem 17.2.1 (C. Loewner). Every Riemannian metric g on thetorus T2 satisfies the inequality

sys1(g)2 ≤ γ2 area(g), (17.2.1)

where γ2 = 2√3is the Hermite constant (17.1.8). A metric attaining

the optimal bound (17.2.1) is necessarily flat, and is homothetic to thequotient of C by the Eisenstein integers, i.e. lattice spanned by the cuberoots of unity, cf. Lemma 15.9.1.


The result can be reformulated in a number of ways.2 Loewner’storus inequality relates the total area, to the systole, i.e. least lengthof a noncontractible loop on the torus (T2, g):

area(g)−√32sys1(g)

2 ≥ 0. (17.2.2)

The boundary case of equality is attained if and only if the metric ishomothetic to the flat metric obtained as the quotient of R2 by thelattice formed by the Eisenstein integers.

17.3. Loewner’s inequality with defect

Loewner’s torus inequality can be strengthened by introducing a“defect” (shegiya, she’erit) term, similar to Bonnesen’s strengtheningof the isoperimetic inequality. To write it down, we need to review theconformal representation theorem (uniformisation theorem).

By the conformal representation theorem, we can assume that themetric g on the torus T2 is of the form

f 2(dx2 + dy2),

with respect to a unit area flat metric dx2+dy2 on the torus T2 viewedas a quotient R2/L of the (x, y) plane by a lattice. The defect termin question is simply the variance (shonut) of the conformal factor fabove. The inequality with the defect term looks as follows.

Theorem 17.3.1. Every metric on the torus satisfies the followingstrengthened form of Loewner’s inequality:

area(g)−√32sys(g)2 ≥ Var(f). (17.3.1)

Here the error term, or isosystolic defect, is given by the variance3

Var(f) =

∫

T2

(f −m)2 (17.3.2)

of the conformal factor f of the metric g = f 2(dx2 + dy2) on the torus,relative to the unit area flat metric g0 = dx2+dy2 in the same conformalclass. Here

m =

∫

T2

fdxdy (17.3.3)

2Thus, in the case of the torus T2, the fundamental group is abelian. Hencethe systole can be expressed in this case as follows: sys1(T

2) = λ1(H1 (T

2;Z), ‖ ‖),

where ‖ ‖ is the stable norm.3shonut

17.5. AN APPLICATION OF THE COMPUTATIONAL FORMULA 181

is the mean4 of f . More concretely, if (T2, g0) = R2/L where L is alattice of unit coarea, and D is a fundamental domain for the actionof L on R2 by translations, then the integral (17.3.3) can be written as

m =

∫

D

f(x, y)dxdy

where dxdy is the standard Lebesgue measure of R2.

17.4. Computational formula for the variance

The proof of inequalities with isosystolic defect relies upon the fa-miliar computational formula for the variance5 of a random variable6

X in terms of expected values.Namely, we have the formula

Eµ(X2)− (Eµ(X))2 = Var(X), (17.4.1)

where µ is a probability measure. Here the variance is

Var(X) = Eµ((X −m)2

),

where m = Eµ(X) is the expected value (i.e. the mean) (tochelet).

17.5. An application of the computational formula

Keeping our differential geometric application in mind, we will de-note the random variable (mishtaneh akra’i) f .

Now consider a flat metric g0 of unit area on the 2-torus T2. Denotethe associated measure by µ. In other words, µ is given by the usualLebesgue measure dxdy. Since µ is a probability measure, we can applyformula (17.4.1) to it. Consider a metric g = f 2g0 conformal to the flatone, with conformal factor f > 0. Then we have

Eµ(f2) =

∫

T2

f 2dxdy = area(g),

since f 2dx dy is precisely the area element of the metric g.Equation (17.4.1) therefore becomes

area(g)− (Eµ(f))2 = Var(f). (17.5.1)

Next, we will relate the expected value Eµ(f) to the systole ofthe metric g. Then we will then relate (17.4.1) to Loewner’s torusinequality.

4tochelet (with “het”)5shonut6mishtaneh mikri


By the uniformisation theorem, every metric on the torus T2 isconformally equivalent to a flat metric of unit area. Denoting such aflat metric by g0, such equivalence can be written as

g = f 2g0.

17.6. Conformal invariant σ

Let us prove Loewner’s torus inequality for the metric g = f 2g0on T2, using the computational formula for the variance. We firstanalyze the expected value term

Eµ(f) =

∫

T2

fdxdy

in (17.5.1).By the proof of Lemma 15.9.1, the lattice of deck transformations

of the flat torus g0 admits a Z-basis similar (domeh) to {τ, 1} ⊆ C,where τ belongs to the standard fundamental domain (15.8.1). In otherwords, the lattice is similar to

Zτ + Z1 ⊆ C.

Definition 17.6.1. We define the conformal invariant σ as follows:consider the imaginary part Im(τ) and set

σ2 := Im(τ) > 0.

Lemma 17.6.2. We have σ2 ≥√32, with equality if and only if τ is

the primitive cube or sixth root of unity.

Proof. This is immediate from the geometry of the fundamentaldomain. �

17.7. Fundamental domain and Loewner’s torus inequality

Thinking of a unit area flat metric g0 as a lattice quotient R2/L, wecan write down the flat metric as dx2+dy2. We think of the conformalfactor f(x, y) > 0 of the metric g on T2 as a doubly periodic (i.e., L-periodic) function on R2.

Theorem 17.7.1. Let σ be the conformal invariant as above. Thenthe metric f 2(dx2 + dy2) satisfies

area(g)− σ2sys(g)2 ≥ Var(f). (17.7.1)

Proof. Since g0 is assumed to be of unit area, the basis for itsgroup of deck tranformations can therefore be taken to be the pair{

τ

σ,1

σ

}.

17.8. BOUNDARY CASE OF EQUALITY 183

Here Im(τσ

)= σ. Thus the torus C/ Span

(τσ, 1σ

)has unit area:

area(C/ Span

(τσ, 1σ

))= 1.

With these normalisations, we see that the flat torus is ruled by a pencilof horizontal closed geodesics, denoted

γy = γy(x),

each of length σ−1, where the “width” of the pencil equals σ, i.e. theparameter y ranges through the interval [0, σ], with γσ = γ0. Note thateach of these closed geodesics is noncontractible in T2, and thereforeby definition

length(γy) ≥ sys1(g).

By Fubini’s theorem, we pass to the iterated integral (nishneh) to ob-tain the following lower bound for the expected value:

Eµ(f) =

∫ σ

0

(∫

γy

f(x)dx

)dy

=

∫ σ

0

length(γy)dy

≥ σsys(g).

See also [Ka07, p. 41, 44]. Substituting into (17.5.1), we obtain therequired inequality

area(g)− σ2sys(g)2 ≥ Var(f). (17.7.2)

where f is the conformal factor of the metric g with respect to the unitarea flat metric g0. �

Since we have in general σ2 ≥√32, we obtain in particular Loewner’s

torus inequality with isosystolic defect,

area(g)−√32sys(g)2 ≥ Var(f). (17.7.3)

cf. [Pu52].

17.8. Boundary case of equality

Corollary 17.8.1. A metric satisfying the boundary case of equal-ity in Loewner’s torus inequality is necessarily flat and homothetic tothe quotient of R2 by the lattice of Eisenstein integers.

Proof. If a metric f 2(dx2 + dy2) satisfies the boundary case of

equality area(g) −√32sys(g)2 = 0, then the variance of the conformal

factor f must vanish by (17.7.3). Hence f is a constant function. The


proof is completed by applying Lemma 15.9.1 on the Hermite constantin dimension 2. �

Now suppose τ is pure imaginary, i.e. the lattice L is a rectangularlattice of coarea 1.

Corollary 17.8.2. If τ is pure imaginary, then the metric g =f 2g0 satisfies the inequality

area(g)− sys(g)2 ≥ Var(f). (17.8.1)

Proof. If τ is pure imaginary then σ =√

Im(τ) ≥ 1, and theinequality follows from (17.7.1). �

In particular, every surface of revolution satisfies (17.8.1), since itslattice is rectangular, cf. Corollary 15.10.1.

17.9. Global geometry of surfaces

Discussion of Local versus Global: The local behavior is by def-inition the behavior in an open neighborhood of a point. The localbehavior of a smooth curve is well understood by the implicit functiontheorem. Namely, a smooth curve in the plane or in 3-space can bethought of as the graph of a smooth function. A curve in the plane islocally the graph of a scalar function. A curve in 3-space is locally thegraph of a vector-valued function.

Example 17.9.1. The unit circle in the plane can be defined im-plicitly by

x2 + y2 = 1,

or parametrically by

t 7→ (cos t, sin t).

Alternatively, it can be given locally as the graph of the function

f(x) =√1− x2.

Note that this presentation works only for points on the upper halfcir-cle. For points on the lower halfcircle we use the function

−√1− x2.

Both of these representations fail at the points (1, 0) and (−1, 0).To overcome this difficulty, we must work with y as the independent

variable, instead of x. Thus, we can parametrize a neighborhood of(1,0) by using the function

x = g(y) =√1− y2.

17.10. DEFINITION OF MANIFOLD 185

Example 17.9.2. The helix given in parametric form by

(x, y, z) = (cos t, sin t, t).

It can also be defined as the graph of the vector-valued function f(z),with values in the (x, y)-plane, where

(x(z), y(z)) = f(z) = (cos z, sin z).

In this case the graph representation in fact works even globally.

Example 17.9.3. The unit sphere in 3-space can be representedlocally as the graph of the function of two variables

f(x, y) =√1− x2 − y2.

As above, concerning the local nature of the presentation necessitatesadditional functions to represent neighborhoods of points not in theopen northern hemisphere (this example is discussed in more detail inSection 17.11).

17.10. Definition of manifold

Motivated by the examples given in the previous section, we give ageneral definition as follows.

A manifold is defined as a subset of Euclidean space which is lo-cally a graph of a function, possibly vector-valued. This is the originaldefinition of Poincare who invented the notion (see Arnold [1, p. 234]).

Definition 17.10.1. By a 2-dimensional closed Riemannian man-ifold we mean a compact subset

M ⊆ Rn

such that in an open neighborhood of every point p ∈ M in Rn, thecompact subsetM can be represented as the graph of a suitable smoothvector-valued function of two variables.

Here the function has values in (n− 2)-dimensional vectors.The usual parametrisation of the graph can then be used to calcu-

late the coefficients gij of the first fundamental form, as, for example,in Theorem 17.21.2 and Example 5.11.1. The collection of all such datais then denoted by the pair (M, g), where

g = (gij)

is referred to as the metric.


Remark 17.10.2. Differential geometers like the (M, g) notation,because it helps separate the topologyM from the geometry g. Strictlyspeaking, the notation is redundant, since the object g already incorpo-rates all the information, including the topology. However, geometershave found it useful to use g when one wants to emphasize the geome-try, and M when one wants to emphasize the topology.

Note that, as far as the intrinsic geometry of a Riemannian manifoldis concerned, the embedding in Rn referred to in Definition 17.10.1is irrelevant to a certain extent, all the more so since certain basicexamples, such as flat tori, are difficult to imbed in a transparent way.

17.11. Sphere as a manifold

The round 2-sphere S2 ⊆ R3 defined by the equation

x2 + y2 + z2 = 1

is a closed Riemannian manifold. Indeed, consider the function f(x, y)

defined in the unit disk x2 + y2 < 1 by setting f(x, y) =√

1− x2 − y2.Define a coordinate chart

x1(u1, u2) = (u1, u2, f(u1, u2)).

Thus, each point of the open northern hemisphere admits a neighbor-hood diffeomorphic to a ball (and hence to R2). To cover the southernhemisphere, use the chart

x2(u1, u2) = (u1, u2,−f(u1, u2)).

To cover the points on the equator, use in addition charts x3(u1, u2) =

(u1, f(u1, u2), u2), x4(u1, u2) = (u1,−f(u1, u2), u2), as well as the pair of

charts x5(u1, u2) = (f(u1, u2), u1, u2), x6(u

1, u2) = (−f(u1, u2), u1, u2).

17.12. Dual bases

Tangent space, cotangent space, and the notation for bases in thesespaces were discussed in Section 14.4.

We will work with dual bases(∂∂ui

)for vectors, and (dui) for cov-

ectors (i.e. elements of the dual space), such that

dui(

∂

∂uj

)= δij , (17.12.1)

where δij is the Kronecker delta.Recall that the metric coefficients are defined by setting

gij = 〈xi, xj〉,

17.13. JACOBIAN MATRIX 187

where x is the parametrisation of the surface. We will only work withmetrics whose first fundamental form is diagonal. We can thus writethe first fundamental form as follows:

g = g11(u1, u2)(du1)2 + g22(u

1, u2)(du2)2. (17.12.2)

Remark 17.12.1. Such data can be computed from a Euclideanembedding as usual, or it can be given apriori without an embedding,as we did in the case of the hyperlobic metric.

We will work with such data independently of any Euclidean em-bedding, as discussed in Section 10.6. For example, if the metric coef-ficients form an identity matrix, we obtain

g =(du1)2

+(du2)2, (17.12.3)

where the interior superscript denotes an index, while exterior super-script denotes the squaring operation.

17.13. Jacobian matrix

The Jacobian matrix of v = v(u) is the matrix

∂(v1, v2)

∂(u1, u2),

which is the matrix of partial derivatives. Denote by

Jacv(u) = det

(∂(v1, v2)

∂(u1, u2)

).

It is shown in advanced calculus that for any function f(v) in a do-main D, one has

∫

D

f(v)dv1dv2 =

∫

D

g(u)Jacv(u)du1du2, (17.13.1)

where g(u) = f(v(u)).

Example 17.13.1. Let u1 = r, u2 = θ. Let v1 = x, v2 = y. Wehave x = r cos θ and y = r sin θ. One easily shows that the Jacobianis Jacv(u) = r. The area elements are related by

dv1dv2 = Jacv(u)du1du2,

or

dxdy = rdrdθ.

A similar relation holds for integrals.


17.14. Area of a surface, independence of partition

Partition7 is what allows us to perform actual calculations witharea, but the result is independent of partition (see below).

Based on the local definition of area discussed in an earlier chapter,we will now deal with the corresponding global invariant.

Definition 17.14.1. The area element dA of the surface is theelement

dA :=√det(gij)du

1du2,

where det(gij) = g11g22 − g212 as usual.

Theorem 17.14.2. Define the area of (M, g) by means of the for-mula

area =

∫

M

dA =∑

{U}

∫

U

√det(gij)du

1du2, (17.14.1)

namely by choosing a partition {U} of M subordinate to a finite opencover as in Definition 17.10.1, performing a separate integration ineach open set, and summing the resulting areas. Then the total area isindependent of the partition and choice of coordinates.

Proof. Consider a change from a coordinate chart (ui) to anothercoordinate chart, denoted (vα). In the overlap of the two domains, thecoordinates can be expressed in terms of each other, e.g. v = v(u), andwe have the 2 by 2 Jacobian matrix Jacv(u).

Denote by gαβ the metric coefficients with respect to the chart (vα).Thus, in the case of a metric induced by a Euclidean embedding definedby x = x(u) = x(u1, u2), we obtain a new parametrisation

y(v) = x(u(v)).

Then we have

gαβ =

⟨∂y

∂vα,∂y

∂vβ

⟩

=

⟨∂x

∂ui∂ui

∂vα,∂x

∂uj∂uj

∂vβ

⟩

=∂ui

∂vα∂uj

∂vβ

⟨∂x

∂ui,∂x

∂uj

⟩

= gij∂ui

∂vα∂uj

∂vβ.

7ritzuf or chaluka?

17.15. CONFORMAL EQUIVALENCE 189

The right hand side is a product of three square matrices:

∂ui

∂vαgij

∂uj

∂vβ.

The matrices on the left and on the right are both Jacobian matrices.Since determinant is multiplicative, we obtain

det(gαβ) = det(gij)det

(∂(u1, u2)

∂(v1, v2)

)2

.

Hence using equation (17.13.1), we can write the area element as

dA = det12 (gαβ)dv

1dv2

= det12 (gαβ) Jacv(u)du

1du2

= det12 (gij) det

(∂(u1, u2)

∂(v1, v2)

)Jacv(u)du

1du2

= det12 (gij) du

1du2

since inverse maps have reciprocal Jacobians by chain rule. Thus theintegrand is unchanged and the area element is well defined. �

17.15. Conformal equivalence

Definition 17.15.1. Two metrics, g = gijduiduj and h = hijdu

iduj,onM are called conformally equivalent , or conformal for short, if thereexists a function f = f(u1, u2) > 0 such that

g = f 2h,

in other words,

gij = f 2hij ∀i, j. (17.15.1)

Definition 17.15.2. The function f above is called the conformalfactor (note that sometimes it is more convenient to refer, instead, tothe function λ = f 2 as the conformal factor).

Theorem 17.15.3. Note that the length of every vector at a givenpoint (u1, u2) is multiplied precisely by f(u1, u2).

Proof. More speficially, a vector v = vi ∂∂ui

which is a unit vectorfor the metric h, is “stretched” by a factor of f , i.e. its length with


respect to g equals f . Indeed, the new length of v is

√g(v, v) = g

(vi

∂

∂ui, vj

∂

∂uj

) 12

=

(g

(∂

∂ui,∂

∂uj

)vivj

) 12

=√gijvivj

=√f 2hijvivj

= f√hijvivj

= f,


Definition 17.15.4. An equivalence class of metrics on M con-formal to each other is called a conformal structure on M (mivnehconformi).

17.16. Geodesic equation

The material in this section has already been dealt with in an earlierchapter.

Perhaps the simplest possible definition of a geodesic β on a surfacein 3-space is in terms of the orthogonality of its second derivative β ′′ tothe surface. The nonlinear second order ordinary differential equationdefining a geodesic is, of course, the “true” if complicated definition.We will now prove the equivalence of the two definitions. Consider aplane curve

Rs

−→α

R2

(u1,u2)

where α = (α1(s), α2(s)). Let x : R2 → R3 be a regular parametrisationof a surface in 3-space. Then the composition

Rs−→α

R2

(u1,u2)−→x

R3

yields a curve

β = x ◦ α.Definition 17.16.1. A curve β = x◦α is a geodesic on the surface x

if one of the following two equivalent conditions is satisfied:

(a) we have for each k = 1, 2,

(αk)′′

+ Γkij(αi)

′

(αj)′

= 0 where′

=d

ds, (17.16.1)

17.17. CLOSED GEODESIC 191

meaning that

(∀k) d2αk

ds2+ Γkij

dαi

ds

dαj

ds= 0;

(b) the vector β ′′ is perpendicular to the surface and one has


′n. (17.16.2)

To prove the equivalence, we write β = x ◦ α, then β ′ = xiαi′ by

chain rule. Furthermore,

β ′′ =d

ds(xi ◦ α)αi′ + xiα

i′′ = xijαj ′αi

′+ xkα

k′′.

Since xij = Γkijxk + Lijn holds, we have

β ′′ − Lijαi′αj

′n = xk

(αk

′′+ Γkijα

i′αj′).

17.17. Closed geodesic

Definition 17.17.1. A closed geodesic in a Riemannian 2-manifoldMis defined equivalently as

(1) a periodic curve β : R → M satisfying the geodesic equa-

tion αk′′+ Γkijα

i′αj′= 0 in every chart x : R2 →M , where, as

usual, β = x◦α and α(s) = (α1(s), α2(s)) where s is arclength.Namely, there exists a period T > 0 such that β(s+T ) = β(s)for all s.

(2) A unit speed map from a circle R/LT → M satisfying thegeodesic equation at each point, where LT = TZ ⊆ R is therank one lattice generated by T > 0.

Definition 17.17.2. The length L(β) of a path β : [a, b] → M iscalculated using the formula

L(β) =

∫ b

a

‖β ′(t)‖dt,

where ‖v‖ =√gijvivj whenever v = vixi. The energy is defined

by E(β) =∫ ba‖β ′(t)‖2dt.

A closed geodesic as in Definition 17.17.1, item 2 has length T .

Remark 17.17.3. The geodesic equation (17.16.1) is the Euler-Lagrange equation of the first variation of arc length. Therefore when apath minimizes arc length among all neighboring paths connecting twofixed points, it must be a geodesic. A corresponding statement is validfor closed loops, cf. proof of Theorem 17.18.1. See also Section 17.1.


In Sections 17.19 and 17.23 we will give a complete description of thegeodesics for the constant curvature sphere, as well as for flat tori.

17.18. Existence of closed geodesic

Theorem 17.18.1. Every free homotopy class of loops in a closedmanifold contains a closed geodesic.

Proof. We sketch a proof for the benefit of a curious reader, whocan also check that the construction is independent of the choices in-volved. The relevant topological notions are defined in Section 18.1and [Hat02]. A free homotopy class α of a manifold M correspondsto a conjugacy class gα ⊂ π1(M). Pick an element g ∈ gα. Thus g actson the universal cover M of M . Let fg : M → R be the displacementfunction of g, i.e.

fg(x) = d(x, g.x).

Let x0 ∈M be a minimum of fg. A first variation argument shows thatany length-minimizing path between x0 and g.x0 descends to a closedgeodesic in M representing α, cf. [Car92, Ch93, GaHL04]. �

17.19. Surfaces of constant curvature

By the uniformisation theorem 14.9.2, all surfaces fall into threetypes, according to whether they are conformally equivalent to metricsthat are:

(1) flat (i.e. have zero Gaussian curvature K ≡ 0);(2) spherical (K ≡ +1);(3) hyperbolic (K ≡ −1).

For closed surfaces, the sign of the Gaussian curvature K is that ofits Euler characteristic, cf. formula (18.9.1).

Theorem 17.19.1 (Constant positive curvature). There are onlytwo compact surfaces, up to isometry, of constant Gaussian curva-ture K = +1. They are the round sphere S2 of Example 17.11; and thereal projective plane, denoted RP2.

17.20. Real projective plane

Intuitively, one thinks of the real projective plane as the quotientsurface obtained if one starts with the northern hemisphere of the 2-sphere, and “glues” together pairs of opposite points of the equatorialcircle (the boundary of the hemisphere).

More formally, the real projective plane can be defined as follows.Let

m : S2 → S2

17.21. SIMPLE LOOPS FOR SURFACES OF POSITIVE CURVATURE 193

denote the antipodal map of the sphere, i.e. the restriction of the map

v 7→ −vin R3. Then m is an involution. In other words, if we consider theaction of the group Z2 = {e,m} on the sphere, each orbit of the Z2

action on S2 consists of a pair of antipodal points

{±p} ⊆ S2. (17.20.1)

Definition 17.20.1. On the set-theoretic level, the real projectiveplane RP2 is the set of orbits of type (17.20.1), i.e. the quotient of S2

by the Z2 action.

Denote by Q : S2 → RP2 the quotient map. The smooth structureand metric on RP2 are induced from S2 in the following sense. Let x :R2 → S2 be a chart on S2 not containing any pair of antipodal points.Let gij be the metric coefficients with respect to this chart.

Let y = m◦x = −x denote the “opposite” chart, and denote by hijits metric coefficients. Then

hij = 〈∂y

∂ui,∂y

∂uj〉 = 〈− ∂x

∂ui,− ∂x

∂uj〉 = 〈 ∂x

∂ui,∂x

∂uj〉 = gij. (17.20.2)

Thus the opposite chart defines the identical metric coefficients. Thecomposition Q ◦ x is a chart on RP2, and the same functions gij formthe metric coefficients for RP2 relative to this chart.

We can summarize the preceeding discussion by means of the fol-lowing definition.

Definition 17.20.2. The real projective plane RP2 is defined inthe following two equivalent ways:

(1) the quotient of the round sphere S2 by (the restriction to S2

of) the antipodal map v 7→ −v in R3. In other words, a typicalpoint of RP2 can be thought of as a pair of opposite points ofthe round sphere.

(2) the northern hemisphere of S2, with opposite points of theequator identified.

The smooth structure of RP2 is induced from the round sphere.Since the antipodal map preserves the metric coefficients by the cal-culation (17.20.2), the metric structure of constant Gaussian curva-ture K = +1 descends to RP2, as well.

17.21. Simple loops for surfaces of positive curvature

Definition 17.21.1. A loop α : S1 → X of a space X is calledsimple if the map α is one-to-one, cf. Definition 18.1.1.


Theorem 17.21.2. The basic properties of the geodesics on surfacesof constant positive curvature as as follows:

(1) all geodesics are closed;(2) the simple closed geodesics on S2 have length 2π and are de-

fined by the great circles;(3) the simple closed geodesics on RP2 have length π;(4) the simple closed geodesics of RP2 are parametrized by half-

great circles on the sphere.

Proof. We calculate the length of the equator of S2. Here thesphere ρ = 1 is parametrized by

x(θ, ϕ) = (sinϕ cos θ, sinϕ sin θ, cosϕ)

in spherical coordinates (θ, ϕ). The equator is the curve x◦α where α(s) =(s, π

2) with s ∈ [0, 2π]. Thus α1(s) = θ(s) = s. Recall that the metric

coefficients are given by g11(θ, ϕ) = sin2 ϕ, while g22 = 1 and g12 = 0.Thus

‖β ′(s)‖ =√gij(s, π

2

)αi′αj ′ =

√(sin π

2

) (dθds

)2= 1.

Thus the length of β is∫ 2π

0

‖β ′(s)‖ds =∫ 2π

0

1ds = 2π.

A geodesic on RP2 is twice as short as on S2, since the antipodal pointsare identified, and therefore the geodesic “closes up” sooner than (i.e.twice as fast as) on the sphere. For example, a longitude of S2 isnot a closed curve, but it descends to a closed curve on RP2, sinceits endpoints (north and south poles) are antipodal, and are thereforeidentified with each other. �

17.22. Successive minima

The material in this section has already been dealt with in an earlierchapter.

Let B be a finite-dimensional Banach space, i.e. a vector space to-gether with a norm ‖ ‖. Let L ⊆ (B, ‖ ‖) be a lattice of maximal rank,i.e. satisfying rank(L) = dim(B). We define the notion of successiveminima of L as follows, cf. [GruL87, p. 58].

Definition 17.22.1. For each k = 1, 2, . . . , rank(L), define the k-thsuccessive minimum λk of the lattice L by

λk(L, ‖ ‖) = inf

{λ ∈ R

∣∣∣∣∃ lin. indep. v1, . . . , vk ∈ Lwith ‖vi‖ ≤ λ for all i

}. (17.22.1)

17.24. HYPERBOLIC SURFACES 195

Thus the first successive minimum, λ1(L, ‖ ‖) is the least length ofa nonzero vector in L.

17.23. Flat surfaces

A metric is called flat if its Gaussian curvature K vanishes at everypoint.

Theorem 17.23.1. A closed surface of constant Gaussian curva-ture K = 0 is topologically either a torus T2 or a Klein bottle.

Let us give a precise description in the former case.

Example 17.23.2 (Flat tori). Every flat torus is isometric to aquotient T2 = R2/L where L is a lattice, cf. [Lo71, Theorem 38.2].In other words, a point of the torus is a coset of the additive ac-tion of the lattice in R2. The smooth structure is inherited from R2.Meanwhile, the additive action of the lattice is isometric. Indeed, wehave dist(p, q) = ‖q − p‖, while for any ℓ ∈ L, we have

dist(p+ ℓ, q + ℓ) = ‖q + ℓ− (p+ ℓ)‖ = ‖q − p‖ = dist(p, q).

Therefore the flat metric on R2 descends to T2.

Note that locally, all flat tori are indistinguishable from the flatplane itself. However, their global geometry depends on the metric in-variants of the lattice, e.g. its successive minima, cf. Definition 17.22.1.Thus, we have the following.

Theorem 17.23.3. The least length of a nontrivial closed geodesicon a flat torus T2 = R2/L equals the first successive minimum λ1(L).

Proof. The geodesics on the torus are the projections of straightlines in R2. In order for a straight line to close up, it must pass througha pair of points x and x+ℓ where ℓ ∈ L. The length of the correspondingclosed geodesic on T2 is precisely ‖ℓ‖, where ‖ ‖ is the Euclidean norm.By choosing a shortest element in the lattice, we obtain a shortestclosed geodesic on the corresponding torus. �

17.24. Hyperbolic surfaces

Most closed surfaces admit neither flat metrics nor metrics of pos-itive curvature, but rather hyperbolic metrics. A hyperbolic surface isa surface equipped with a metric of constant Gaussian curvature K =−1. This case is far richer than the other two.


Example 17.24.1. The pseudosphere (so called because its Gauss-ian curvature is constant, and equals −1) is the surface of revolution

(f(ϕ) cos θ, f(ϕ) sin θ, g(ϕ))

in R3 defined by the functions f(φ) = eφ and

g(φ) =

∫ φ

0

(1− e2ψ

)1/2dψ,

where φ ranges through the interval −∞ < φ ≤ 0. The usual formu-

las g11 = f 2 as well as g22 =(drdφ

)2+(dgdφ

)2yield in our case g11 = e2φ,

while

g22 =(eφ)2 +

(√1− e2φ

)2

= e2φ + 1− e2φ

= 1.

Thus (gij) =

(e2φ 00 1

). The pseudosphere has constant Gaussian

curvature −1, but it is not a closed surface (as it is unbounded in R3).

17.25. Hyperbolic plane

This was already discussed in Section 12.5. The metric

gH2 =1

y2(dx2 + dy2) (17.25.1)

in the upperhalf plane

H2 = {(x, y) | y > 0}is called the hyperbolic metric of the upper half plane.

Theorem 17.25.1. The metric (17.25.1) has constant Gaussiancurvature K = −1.

Proof. By Theorem 12.4.2, we have

K = −∆LB ln f = ∆LB ln y = y2(− 1

y2

)= −1,

as required. �

In coordinates (u1, u2), we can write it, a bit awkwardly, as

gH2 =1

(u2)2

((du1)2

+(du2)2)

.

The Riemannian manifold (H2, gH2) is referred to as the Poincare up-perhalf plane. Its significance resides in the following theorem.

17.25. HYPERBOLIC PLANE 197

Theorem 17.25.2. Every closed hyperbolic surface M is isomet-ric to the quotient of the Poincare upperhalf plane by the action of asuitable group Γ:

M = H2/Γ.

Here the nonabelian group Γ is a discrete subgroup Γ ⊆ PSL(2,R),

where a matrix A =

(a bc d

)acts on H2 = {z ∈ C|ℑ(z) > 0} by

z 7→ az + b

cz + d,

called fractional linear transformations, of Mobius transformations. Allsuch transformations are isometries of the hyperbolic metric. The fol-lowing theorem is proved, for example, in [Kato92].

Theorem 17.25.3. Every geodesic in the Poincare upperhalf planeis either a vertical ray, or a semicircle perpendicular to the x-axis.

The foundational significance of this model in the context of theparallel postulate of Euclid has been discussed by numerous authors.

Example 17.25.4. The length of a vertical interval joining i to cican be calculated as follows. Recall that the conformal factor is f(x, y) =1y. The length is therefore given by

∣∣∣∣∫ c

1

1

ydy

∣∣∣∣ = | ln c|.

Here the substitution y = es gives an arclength parametrisation.

CHAPTER 18

Elements of the topology of surfaces

18.1. Loops, simply connected spaces

We would like to provide a self-contained explanation of the topo-logical ingredient which is necessary so as to understand Loewner’storus inequality, i.e. essentially the notion of a noncontractible loopand the fundamental group of a topological space X . See [Hat02,Chapter 1] for a more detailed account.

Definition 18.1.1. A loop in X can be defined in one of two equiv-alent ways:

(1) a continuous map β : [a, b] → X satisfying β(a) = β(b);(2) a continuous map λ : S1 → X from the circle S1 to X .

Lemma 18.1.2. The two definitions of a loop are equivalent.

Proof. Consider the unique increasing linear function

f : [a, b] → [0, 2π]

which is one-to-one and onto. Thus, f(t) = 2π(t−a)b−a . Given a map

λ(eis) : S1 → X,

we associate to it a map β(t) = λ(eif(t)

), and vice versa. �

Definition 18.1.3. A loop S1 → X is said to be contractible ifthe map of the circle can be extended to a continuous map of the unitdisk D → X , where S1 = ∂D.

Definition 18.1.4. A space X is called simply connected if everyloop in X is contractible.

Theorem 18.1.5. The sphere Sn ⊆ Rn+1 which is the solution setof x20+ . . .+x2n = 1 is simply connected for n ≥ 2. The circle S1 is notsimply connected.

18.2. Orientation on loops and surfaces

Let S1 ⊆ C be the unit circle. The choice of an orientation on thecircle is an arrow pointing clockwise or counterclockwise. The standard

199

200 18. ELEMENTS OF THE TOPOLOGY OF SURFACES

choice is to consider S1 as an oriented manifold with orientation chosencounterclockwise.

If a surface is embedded in 3-space, one can choose a continuousunit normal vector n at every point. Then an orientation is defined bythe right hand rule with respect to n thought of as the thumb (agudal).

18.3. Cycles and boundaries

The singular homology groups with integer coefficients, Hk(M ;Z)for k = 0, 1, . . . ofM are abelian groups which are homotopy invariantsof M . Developing the singular homology theory is time-consuming.The case that we will be primarily interested in as far as these notesare concerned, is that of the 1-dimensional homology group:

H1(M ;Z).

In this case, the homology groups can be characterized easily withoutthe general machinery of singular simplices.

Let S1 ⊆ C be the unit circle, which we think of as a 1-dimensionalmanifold with an orientation given by the counterclockwise direction.

Definition 18.3.1. A 1-cycle α on a manifold M is an integerlinear combination

α =∑

i

nifi

where ni ∈ Z is called the multiplicity (ribui), while each

fi : S1 →M

is a loop given by a smooth map from the circle to M , and each loopis endowed with the orientation coming from S1.

Definition 18.3.2. The space of 1-cycles on M is denoted

Z1(M ;Z).

Let (Σg, ∂Σg) be a surface with boundary ∂Σg , where the genus gis irrelevant for the moment and is only added so as to avoid confusionwith the summation symbol

∑.

The boundary ∂Σg is a disjoint union of circles. Now assume thesurface Σg is oriented.

Proposition 18.3.3. The orientation of the surface induces anorientation on each boundary circle.

Thus we obtain an orientation-preserving identification of each bound-ary component with the standard unit circle S1 ⊆ C (with its counter-clockwise orientation).

18.4. FIRST SINGULAR HOMOLOGY GROUP 201

Given a map Σg → M , its restriction to the boundary thereforeproduces a 1-cycle

∂Σg ∈ Z1(M ;Z).

Definition 18.3.4. The space

B1(M ;Z) ⊆ Z1(M ;Z)

of 1-boundaries in M is the space of all cycles∑

i

nifi ∈ Z1(M ;Z)

such that there exists a map of an oriented surface Σg →M (for some g)satisfying

∂Σg =∑

i

nifi.

Example 18.3.5. Consider the cylinder

x2 + y2 = 1, 0 ≤ z ≤ 1

of unit height. The two boundary components correspond to the twocircles: the “bottom” circle Cbottom defined by z = 0, and the “top”circle Ctop defined by and z = 1. Consider the orientation on thecylinder defined by the outward pointing normal vector. It induces thecounterclockwise orientation on Cbottom, and a clockwise orientationon Ctop.

Now let C0 and C1 be the same circles with the following choice oforientation: we choose a standard counterclockwise parametrisation onboth circles, i.e., parametrize them by means of (cos θ, sin θ). Then theboundary of the cylinder is the difference of the two circles: C0 − C1,or C1 − C0, depending on the choice of orientation.

Example 18.3.6. Cutting up a circle of genus 2 into two once-holedtori shows that the separating curve is a 1-boundary.

Theorem 18.3.7. On a closed orientable surface, a separating curveis a boundary, while a non-separating loop is never a boundary.

18.4. First singular homology group

Definition 18.4.1. The 1-dimensional homology group of M withinteger coefficients is the quotient group

H1(M ;Z) = Z1(M ;Z)/B1(M ;Z).

Definition 18.4.2. Given a cycle C ∈ Z1(M ;Z), its homologyclass will be denoted [C] ∈ H1(M ;Z).


Example 18.4.3. A non-separating loop on a closed surface repre-sents a non-trivial homology class of the surface.

Theorem 18.4.4. The 1-dimensional homology group H1(M ;Z) isthe abelianisation of the fundamental group π1(M):

H1(M ;Z) = (π1M)ab .

Note that a significant difference between the fundamental groupand the first homology group is the following. While only based loopsparticipate in the definition of the fundamental group, the definitionof H1(M ;Z) involves free (not based) loops.

Example 18.4.5. The fundamental groups of the real projectiveplane RP2 and the 2-torus T2 are already abelian. Therefore one ob-tains

H1(RP2;Z) = Z/2Z,and

H1(T2;Z) = Z2.

Example 18.4.6. The fundamental group of an orientable closedsurface Σg of genus g is known to be a group on 2g generators with asingle relation which is a product of g commutators. Therefore one has

H1(Σg;Z) = Z2g.

18.5. Stable norm in 1-dimensional homology

Assume the manifoldM has a Riemannian metric. Given a smoothloop f : S1 → M , we can measure its volume (length) with respect tothe metric of M . We will denote this length by

vol(f)

with a view to higher-dimensional generalisation.

Definition 18.5.1. The volume (length) of a 1-cycle C =∑

i nifiis defined as

vol(C) =∑

i

|ni| vol(fi).

Definition 18.5.2. Let α ∈ H1(M ;Z) be a 1-dimensional homol-ogy class. We define the volume of α as the infimum of volumes ofrepresentative 1-cycles:

vol(α) = inf {vol(C) | C ∈ α} ,where the infimum is over all cycles C =

∑i nifi representing the

class α ∈ H1(M ;Z).

18.6. THE DEGREE OF A MAP 203

The following phenomenon occurs for orientable surfaces.

Theorem 18.5.3. LetM be an orientable surface, i.e. 2-dimensionalmanifold. Let α ∈ H1(M ;Z). For all j ∈ N, we have

vol(jα) = j vol(α), (18.5.1)

where jα denotes the class α+ α + . . .+ α, with j summands.

Proof. To fix ideas, let j = 2. By Lemma 18.5.4 below, a mini-mizing loop C representing a multiple class 2α will necessarily intersectitself in a suitable point p. Then the 1-cycle represented by C can bedecomposed into the sum of two 1-cycles (where each can be thoughtof as a loop based at p). The shorter of the two will then give a min-imizing loop in the class α which proves the identity (18.5.1) in thiscase. The general case follows similarly. �

Lemma 18.5.4. A loop going around a cylinder twice necessarilyhas a point of self-intersection.

Proof. We think of the loop as the graph of a 4π-periodic func-tion f(t) (or alternatively a function on [0, 4π] with equal values at theendpoints). Consider the difference g(t) = f(t) − f(t + 2π). Then gtakes both positive and negative values. By the intermediate functiontheorem, the function g must have a zero t0. Then f(t0) = f(t0 + 2π)hence t0 is a point of self-intersection of the loop. �

18.6. The degree of a map

An example of a degree d map is most easily produced in the caseof a circle. A self-map of a circle given by

eiθ 7→ eidθ

has degree d.We will discuss the degree in the context of surfaces only.

Definition 18.6.1. The degree

df

of a map f between closed surfaces is the algebraic number of pointsin the inverse image of a generic point of the target surface.

We can use the 2-dimensional homology groups defined elsewherein these notes, so as to calculate the degree as follows. Recall that

H2(M ;Z) = Z,

where the generator is represented by the identity self-map of the sur-face.


Theorem 18.6.2. A map

f :M1 →M2

induces a homomorphism

f∗ : H2(M1;Z) → H2(M2;Z),

corresponding to multiplication by the degree df once the groups areidentified with Z:

Z → Z, n 7→ df n.

18.7. Degree of normal map of an embedded surface

Theorem 18.7.1. Let Σ ⊆ R3 be an embedded surface. Let p be itsgenus. Consider the normal map

fn : Σ → S2

defined by sending each point x ∈ Σ to the normal vector n = nx at x.Then the degree of the normal map is precisely 1− p.

Example 18.7.2. For the unit sphere, the normal map is the iden-tity map. The genus is 0, while the degree of the normal map is 1−p =1.

Example 18.7.3. For the torus, the normal map is harder to visu-alize. The genus is 1, while the degree of the normal map is 0.

Example 18.7.4. For a genus 2 surface embedded in R3, the degreeof the normal map is 1 − 2 = −1. This means that if the surface isoriented by the outward-pointing normal vector, the normal map isorientation-reversing.

18.8. Euler characteristic of an orientable surface

The Euler characteristic χ(M) is even for closed orientable surfaces,and the integer p = p(M) ≥ 0 defined by

χ(M) = 2− 2p

is called the genus of M .

Example 18.8.1. We have p(S2) = 0, while p(T2) = 1.

In general, the genus can be understood intuitively as the number of“holes”, i.e. “handles”, in a familiar 3-dimensional picture of a pretzel.We see from formula (18.9.1) that the only compact orientable surfaceadmitting flat metrics is the 2-torus. See [Ar83] for a friendly topo-logical introduction to surfaces, and [Hat02] for a general definition ofthe Euler characteristic.

18.10. CHANGE OF METRIC EXPLOITING GAUSSIAN CURVATURE 205

18.9. Gauss-Bonnet theorem

Every embedded closed surface in 3-space admits a continuous choiceof a unit normal vector n = nx at every point x. Note that no suchchoice is possible for an embedding of the Mobius band, see [Ar83] formore details on orientability and embeddings.

Closed embedded surfaces in R3 are called orientable.

Remark 18.9.1. The integrals of type∫

M

will be understood in the sense of Theorem 17.14.2, namely using animplied partition subordinate to an atlas, and calculating the integralusing coordinates (u1, u2) in each chart, so that we can express themetric in terms of metric coefficients

gij = gij(u1, u2)

and similarly the Gaussian curvature

K = K(u1, u2).

Theorem 18.9.2 (Gauss-Bonnet theorem). Every closed surfaceMsatisfies the identity

∫

M

K(u1, u2)√

det(gij)du1du2 = 2πχ(M), (18.9.1)

where K is the Gaussian curvature function on M , whereas χ(M) isits Euler characteristic.

18.10. Change of metric exploiting Gaussian curvature

We will use the term pseudometric for a quadratic form (or theassociated bilinear form), possibly degenerate.

We would like to give an indication of a proof of the Gauss-Bonnettheorem. We will have to avoid discussing some technical points. Con-sider the normal map

F :M → S2, x 7→ nx.

Consider a neigborhood in M where the map F is a homeomorphism(this is not always possible, and is one of the technical points we areavoiding).

Definition 18.10.1. Let gM the metric of M , and h the standardmetric of S2.


Given a point x ∈M in such a neighborhood, we can calculate thecurvature K(x). We can then consider a new metric in the conformalclass of the metric gM , defined as follows.

Definition 18.10.2. We define a new pseudometric, denoted gM ,on M by multiplying by the conformal factor K(x) at the point x.Namely, gM is the pseudometric which at the point x is given by thequadratic form

gx = K(x)gx.

If K ≥ 0 then the length of vectors is multiplied by√K.

The key to understanding the proof of the Gauss-Bonnet theoremin the case of embedded surfaces is the following theorem.

Theorem 18.10.3. Consider the restriction of the normal map Fto a neighborhood as above. We modify the metric on the source M bythe conformal factor given by the Gaussian curvature, as above. Thenthe map

F : (M, gM) → (S2, h)

preserves areas: the area of the neighborhood in M (with respect to themodified metric) equals to the “area” of its image on the sphere.

18.11. Gauss map

Definition 18.11.1. The Gauss map is the map

F :M → S2, p 7→ Np

defined by sending a point p of M the unit normal vector N = Np

thought of as a point of S2.

The map F sends an infinitesimal parallelogram on the surface, toan infinitesimal parallelogram on the sphere.

We may identify the tangent space to M at p and the tangentspace to S2 at F (p) ∈ S2. Then the differential of the map F is theWeingarten map

W : TpM → TF (p)S2.

The element of area KdA of the surface is mapped to the element ofarea of the sphere. In other words, we modify the element of areaby multiplying by the determinant (Jacobian) of the Weingarten map,namely the Gaussian curvature K(p). Hence the image of the areaelement KdA is precisely area 2-form h on the sphere, as discussed inthe previous section.

It remains to be checked that the map has topological degree givenby half the Euler characteristic of the surface M , proving the theo-rem in the case of embedded surfaces. Since degree is invariant under

18.12. AN IDENTITY 207

continuous deformations, the result can be checked for a particularstandard embedding of a surface of arbitrary genus in R3.

18.12. An identity

Another way of writing identity (18.5.1) is as follows:

vol(α) =1

jvol(jα).

This phenomenon is no longer true for higher-dimensional mani-folds. Namely, the volume of a homology class is no longer multiplica-tive. However, the limit as j → ∞ exists and is called the stable norm.

Definition 18.12.1. Let M be a compact manifold of arbitrarydimension. The stable norm ‖ ‖ of a class α ∈ H1(M ;Z) is the limit

‖α‖ = limj→∞

1

jvol(jα). (18.12.1)

It is obvious from the definition that one has ‖α‖ ≤ vol(α). How-ever, the inequality may be strict in general. As noted above, for2-dimensional manifolds we have ‖α‖ = vol(α).

Proposition 18.12.2. The stable norm vanishes for a class of finiteorder.

Proof. If α ∈ H1(M,Z) is a class of finite order, one has finitelymany possibilities for vol(jα) as j varies. The factor of 1

jin (18.12.1)

shows that ‖α‖ = 0. �

Similarly, if two classes differ by a class of finite order, their stablenorms coincide. Thus the stable norm passes to the quotient latticedefined below.

Definition 18.12.3. The torsion subgroup of H1(M ;Z) will bedenoted T1(M) ⊂ H1(M ;Z). The quotient lattice L1(M) is the lattice

L1(M) = H1(M ;Z)/T1(M).

Proposition 18.12.4. The lattice L1(M) is isomorphic to Zb1(M),where b1 is called the first Betti number of M .

Proof. This is a general result in the theory of finitely generatedabelian groups. �


18.13. Stable systole

Definition 18.13.1. Let M be a manifold endowed with a Rie-mannian metric, and consider the associated stable norm ‖ ‖. Thestable 1-systole of M , denoted stsys1(M), is the least norm of a 1-homology class of infinite order:

stsys1(M) = inf{‖α‖ | α ∈ H1 (M,Z) \ T1(M)

}

= λ1(L1(M), ‖ ‖

).

Example 18.13.2. For an arbitrary metric on the 2-torus T2, the1-systole and the stable 1-systole coincide by Theorem 18.5.3:

sys1(T2) = stsys1(T

2),

for every metric on T2.

18.14. Free loops, based loops, and fundamental group

One can refine the notion of simple connectivity by introducing agroup, denoted

π1(X) = π1(X, x0),

and called the fundamental group of X relative to a fixed “base”point x0 ∈ X .

Definition 18.14.1. A based loop is a loop α : [0, 1] → X satisfyingthe condition α(0) = α(1) = x0.

In terms of the second item of Definition 18.1.1, we choose a fixedpoint s0 ∈ S1. For example, we can choose s0 = ei0 = 1 for theusual unit circle S1 ⊆ C, and require that α(s0) = x0. Then thegroup π1(X) is the quotient of the space of all based loops modulothe equivalence equivalence relation defined by homotopies fixing thebasepoint, cf. Definition 18.14.2.

The equivalence class of the identity element is precisely the setof contractible loops based at x0. The equivalence relation can bedescribed as follows for a pair of loops in terms of the second item ofDefinition 18.1.1.

Definition 18.14.2. Two based loops α, β : S1 → X are equiva-lent, or homotopic, if there is a continuous map of the cylinder S1 ×[c, d] → X whose restriction to S1×{c} is α, whose restriction to S1×{d} is β, while the map is constant on the segment {s0} × [c, d] of thecylinder, i.e. the basepoint does not move during the homotopy.

18.15. FUNDAMENTAL GROUPS OF SURFACES 209

Definition 18.14.3. An equivalence class of based loops is calleda based homotopy class. Removing the basepoint restriction (as well asthe constancy condition of Definition 18.14.2), we obtain a larger classcalled a free homotopy class (of loops).

Definition 18.14.4. Composition of two loops is defined most con-veniently in terms of item 1 of Definition 18.1.1, by concatenating theirdomains.

In more detail, the product of a pair of loops, α : [−1, 0] → Xand β : [0,+1] → X , is a loop α.β : [−1, 1] → X , which coincideswith α and β in their domains of definition. The product loop α.βis continuous since α(0) = β(0) = x0. Then Theorem 18.1.5 can berefined as follows.

18.15. Fundamental groups of surfaces

Theorem 18.15.1. We have π1(S1) = Z, while π1(Sn) is the trivial

group for all n ≥ 2.

Definition 18.15.2. The 2-torus T2 is defined to be the followingCartesian product: T2 = S1 × S1, and can thus be realized as a subset

T2 = S1 × S1 ⊆ R2 × R2 = R4.

We have π1(T2) = Z2. The familiar doughnut picture realizes T2 asa subset in Euclidean space R3.

Definition 18.15.3. A 2-dimensional closed Riemannian manifold(i.e. surface) M is called orientable if it can be realized by a subsetof R3.

Theorem 18.15.4. The fundamental group of a surface of genus gis isomorphic to a group on 2g generators a1, b1, . . . , ag, bg satisfyingthe unique relation

g∏

i=1

aibia−1i b−1

i = 1.

Note that this is not the only possible presentation of the group interms of a single relation.

CHAPTER 19

Pu’s inequality

19.1. Hopf fibration h

To prove Pu’s inequality, we will need to study the Hopf fibrationmore closely.

The circle action in C2 restricts to the unit sphere S3 ⊂ C2, whichtherefore admits a fixed point free circle action. Namely, the circle

S1 = {eiθ : θ ∈ R} ⊆ C

acts on a point (z1, z2) ∈ C2 by

eiθ.(z1, z2) = (eiθz1, eiθz2). (19.1.1)

The quotient manifold (space of orbits) S3/S1 is the 2-sphere S2 (fromthe projective viewpoint, the quotient space is the complex projectiveline CP1).

Definition 19.1.1. The quotient map

h : S3 → S2, (19.1.2)

called the Hopf fibration.

19.2. Tangent map

Proposition 19.2.1. Consider a smooth map f :M → N betweendifferentiable manifolds. Then f defines a natural map

df : TM → TN

called the tangent map.

Proof. We represent a tangent vector v ∈ TpM by a curve

c(t) : I →M,

such that c′(o) = v. The composite map σ = f ◦ c : I → N is a curvein N . Then the vector σ′(0) is the image of v under df :

df(v) = σ′(0)

by chain rule, proving the proposition. �

211

212 19. PU’S INEQUALITY

19.3. Riemannian submersion

Definition 19.3.1. A projection (R2, g) → R to the u1-axis iscalled a Riemannian submersion if the metric coefficients in the hori-zontal direction are independent of u2, or more precisely,

g = h(u1)(du1)2

+ k(u1, u2)(du2)2, (19.3.1)

for suitable functions h and k (note that the matrix of metric coeffi-cients is diagonal but not necessarily scalar).

Replacing the coordinates by (x, y), we obtain a somewhat morereadable formula

g = h(x)dx2 + k(x, y)dy2.

Being a Riemannian submersion is, of course, a local property, and westated it for Euclidean space for convenience. We will use it mostly inthe context of maps between compact manifolds.

Example 19.3.2. The metric (17.25.1) of the Poincare upperhalfplane admits a Riemannian submersion to (the positive ray of) the y-axis but not to the x-axis, as the conformal factor depends on y.

19.4. Riemannian submersions

Riemannian submersions in the 2-dimensional case were dealt within an Section 19.3.

Consider a fiber bundle map f :M → N between closed manifolds,i.e., a smooth map whose df is onto. Let F ⊆ M be the fiber overa point p ∈ N . The differential df : TM → TN vanishes on thesubspace TxF ⊆ TM at every point x ∈ F . More precisely, we havethe vertical space1

ker(dfx) = TxF.

Given a Riemannian metric on M , we can consider the orthogonalcomplement of TxF in TxM , denoted

Hx ⊆ TxM.

ThusTxM = TxF ⊕Hx

is an orthogonal decomposition.

Definition 19.4.1. A fiber bundle map f : M → N betweenRiemannian manifolds is called a Riemannian submersion if at everypoint x ∈ M , the restriction of df : Hx → Tf(x)N is an isometry.

1anchi

19.5. HAMILTON QUATERNIONS 213

Proposition 19.4.2. Let S3 be the 3-sphere of constant sectionalcurvature +1, i.e., the unit sphere in R4. The Hopf fibration

h : S3 → S2

is a Riemannian submersion, for which the natural metric on the base S2

is a metric of constant Gaussian curvature +4.

The latter is explained by the fact that the maximal distance be-tween a pair of S1 orbits is π

2rather than π, in view of the fact that

a pair of antipodal points of S3 lies in a common orbit and thereforedescends to the same point of the quotient space.

A pair of points at maximal distance is defined by a pair v, w ∈ S3

such that w is orthogonal to Cv.

19.5. Hamilton quaternions

The quaternionic viewpoint will be applied in Section ?The algebraof the Hamilton quaternions is the real 4-dimensional vector space withreal basis {1, i, j, k}, so that

H = R1 + Ri+ Rj + Rk

equipped with an associative (chok kibutz), non-commutative productoperation. This operation has the following properties:

(1) the center of H is R1;(2) the operation satisfies the relations i2 = j2 = k2 = −1 and ij =

−ji = k, jk = −kj = i, ki = −ik = j.

Lemma 19.5.1. The algebra H is naturally isomorphic to the com-plex vector space C2 via the identification R1 + Ri ≃ C.

Proof. The subspace Rj + Rk can be thought of as

Rj + Rij = (R1 + Ri)j,

and we therefore obtain a natural isomorphism

H = C1 + Cj,

showing that {1, j} is a complex basis for H. �

Theorem 19.5.2. Nonzero quaternions form a group under quater-nionic multiplication.

Proof. Given q = a+bi+cj+dk ∈ H, let N(q) = a2+b2+c2+d2,and q = a − bi − cj − dk. Then qq = N(q). Therefore we have amultiplicative inverse

q−1 =1

N(q)q,


214 19. PU’S INEQUALITY

19.6. Complex structures on the algebra H

CHAPTER 20

Approach using energy-area identity

20.1. An integral-geometric identity

Let T2 be a torus with an arbitrary metric. Let T0 = R2/L be theflat torus conformally equivalent to T2. Let ℓ0 = ℓ0(x) be a simple

closed geodesic of T0. Thus ℓ0 is the projection of a line ℓ0 ⊆ R2.Let ℓy ⊆ R2 be the line parallel to ℓ0 at distance y > 0 from ℓ0 (here

we must “choose sides”, e.g. by orienting ℓ0 and requiring ℓy to lie to

the left of ℓ0). Denote by ℓy ⊆ T0 the closed geodesic loop defined by

the image of ℓy. Let y0 > 0 be the smallest number such that ℓy0 = ℓ0,

i.e. the lines ℓy0 and ℓ0 both project to ℓ0.Note that the loops in the family {ℓy} ⊆ T2 are not necessarily

geodesics with respect to the metric of T2. On the other hand, thefamily satisfies the following identity.

Lemma 20.1.1 (An elementary integral-geometric identity). Themetric on T2 satisfies the following identity:

area(T2) =

∫ y0

0

E(ℓy)dy, (20.1.1)

where E is the energy of a loop with respect to the metric of T2, seeDefinition 17.17.2.

Proof. Denote by f 2 the conformal factor of T2 with respect tothe flat metric T0. Thus the metric on T2 can be written as

f 2(x, y)(dx2 + dy2).

By Fubini’s theorem applied to a rectangle with sides lengthT0(ℓ0)

and y0, combined with Theorem 17.15.3, we obtain

area(T2) =

∫

T0

f 2dxdy

=

∫ y0

0

(∫

ℓy

f 2(x, y)dx

)dy

=

∫ y0

0

E(ℓy)dy,

215

216 20. APPROACH USING ENERGY-AREA IDENTITY


Remark 20.1.2. The identity (20.1.1) can be thought of as thesimplest integral-geometric identity, cf. equation (20.4.3).

20.2. Two proofs of the Loewner inequality

We give a slightly modified version of M. Gromov’s proof [Gro96],using conformal representation and the Cauchy-Schwarz inequality, ofthe Loewner inequality (17.2.1) for the 2-torus, see also [CK03]. Wepresent the following slight generalisation: there exists a pair of closedgeodesics on (T2, g), of respective lengths λ1 and λ2, such that

λ1λ2 ≤ γ2 area(g), (20.2.1)

and whose homotopy classes form a generating set for π1(T2) = Z×Z.

Proof. The proof relies on the conformal representation

φ : T0 → (T2, g),

where T0 is flat, cf. uniformisation theorem 14.9.2. Here the map φmay be chosen in such a way that (T2, g) and T0 have the same area.Let f be the conformal factor, so that

g = f 2((du1)2 + (du2)2

)

as in formula (17.15.1), where (du1)2+(du2)2 (locally) is the flat metric.Let ℓ0 be any closed geodesic in T0. Let {ℓs} be the family of

geodesics parallel to ℓ0. Parametrize the family {ℓs} by a circle S1 oflength σ, so that

σℓ0 = area(T0).

Thus T0 → S1 is a Riemannian submersion, cf. Definition 19.3.1. Then

area(T2) =

∫

T0

f 2.

By Fubini’s theorem, we have area(T2) =∫S1 ds

∫ℓsf 2dt. Therefore by

the Cauchy-Schwarz inequality,

area(T2) ≥∫

S1

ds

(∫ℓsfdt)2

ℓ0=

1

ℓ0

∫

S1

ds (lengthφ(ℓs))2 .

Hence there is an s0 such that area(T2) ≥ σℓ0lengthφ(ℓs0)

2, so that

lengthφ(ℓs0) ≤ ℓ0.

This reduces the proof to the flat case. Given a lattice in C, we choosea shortest lattice vector λ1, as well as a shortest one λ2 not proportionalto λ1. The inequality now follows from Lemma 15.9.1. In the boundary

20.2. TWO PROOFS OF THE LOEWNER INEQUALITY 217

case of equality, one can exploit the equality in the Cauchy-Schwarzinequality to prove that the conformal factor must be constant. �

Alternative proof. Let ℓ0 be any simple closed geodesic in T0.Since the desired inequality (20.2.1) is scale-invariant, we can assumethat the loop has unit length:

lengthT0(ℓ0) = 1,

and, moreover, that the corresponding covering transformation of theuniversal cover C = T0 is translation by the element 1 ∈ C. Wecomplete 1 to a basis {τ, 1} for the lattice of covering transformationsof T0. Note that the rectangle defined by

{z = x+ iy ∈ C | 0 < x < 1, 0 < y < Im(τ)]}is a fundamental domain for T0, so that area(T0) = Im(τ). The maps

ℓy(x) = x+ iy, x ∈ [0, 1]

parametrize the family of geodesics parallel to ℓ0 on T0. Recall thatthe metric of the torus T is f 2(dx2 + dy2).

Lemma 20.2.1. We have the following relation between the lengthand energy of a loop:

length(ℓy)2 ≤ E(ℓy).

Proof. By the Cauchy-Schwarz inequality,

∫ 1

0

f 2(x, y)dx ≥(∫ 1

0

f(x, y)dx

)2

,


Now by Lemma 20.1.1 and Lemma 20.2.1, we have

area(T2) ≥∫ Im(τ)

0

(length(ℓy)

)2dy.

Hence there is a y0 such that

area(T2) ≥ Im(τ) length(ℓy0)2,

so that length(ℓy0) ≤ 1. This reduces the proof to the flat case. Givena lattice in C, we choose a shortest lattice vector λ1, as well as ashortest one λ2 not proportional to λ1. The inequality now followsfrom Lemma 15.9.1. �


Remark 20.2.2. Define a conformal invariant called the capacityof an annulus as follows. Consider a right circular cylinder

ζκ = R/Z× [0, κ]

based on a unit circle R/Z. Its capacity C(ζκ) is defined to be itsheight, C(ζκ) = κ. Recall that every annular region in the plane is con-formally equivalent to such a cylinder, and therefore we have defined aconformal invariant of an arbitrary annular region. Every annular re-gion R satisfies the inequality area(R) ≥ C(R) sys1(R)

2. Meanwhile, ifwe cut a flat torus along a shortest loop, we obtain an annular region R

with capacity at least C(R) ≥ γ−12 =

√32. This provides an alternative

proof of the Loewner theorem. In fact, we have the following identity:

confsys1(g)2C(g) = 1, (20.2.2)

where confsys is the conformally invariant generalisation of the homol-ogy systole, while C(g) is the largest capacity of a cylinder obtainedby cutting open the underlying conformal structure on the torus.

Question 20.2.3. Is the Loewner inequality (17.2.1) satisfied byevery orientable nonsimply connected compact surface? Inspite of itselementary nature, and considerable research devoted to the area, thequestion is still open. Recently the case of genus 2 was settled as wellas genus g ≥ 20

20.3. Hopf fibration and the Hamilton quaternions

The circle action in C2 restricts to the unit sphere S3 ⊂ C2, whichtherefore admits a fixed point free circle action. Namely, the circle S1 ={eiθ : θ ∈ R} ⊆ C acts on (z1, z2) by

eiθ.(z1, z2) = (eiθz1, eiθz2).

It is possible to show that the quotient manifold (space of orbits) S3/S1

is the 2-sphere S2 (from the projective viewpoint, the quotient is thecomplex projective line CP1). Moreover, the quotient map

S3 → S2, (20.3.1)

called the Hopf fibration, is a Riemannian submersion, for which thenatural metric on the base is a metric of constant Gaussian curva-ture +4 rather than +1. The latter is explained by the fact that themaximal distance between a pair of S1 orbits is π

2rather than π, in

view of the fact that a pair of antipodal points of S3 lies in a commonorbit.

Furthermore, the sphere S3 can be thought of as the Lie groupformed of the unit (Hamilton) quaternions inH = C2. The Lie group SO(3)

20.4. DOUBLE FIBRATION OF SO(3) AND INTEGRAL GEOMETRY ON S2219

can be identified with S3/{±1}. Therefore the fibration (20.3.1) de-scends to a fibration

SO(3) → S2, (20.3.2)

exploited in the next section.

Remark 20.3.1. The group G = SO(3) can be identified with theunit tangent bundle of S2, as follows. Choose a fixed unit tangentvector v to S2, and send an element g ∈ G, acting on S2, to the imageof v under dg : TS2 → TS2.

Remark 20.3.2. Let us elaborate a bit further on the quaternionicviewpoint, with an eye to aplying it in Section 20.4. A choice of a purelyimaginary Hamilton quaternion in H = R1+Ri+Rj+Rk, i.e. a linearcombination of i, j, and k, specifies a complex structure on H. Such acomplex structure leads to a fibration of the unit sphere, as in (20.3.1).Choosing a pair of purely imaginary quaternions which are orthogonal,results in a pair of circle fibrations of S3 such that a fiber of one fibrationis perpendicular to a fiber of the other fibration whenever two suchfibers meet. The construction of two “perpendicular” fibrations alsodescends to the quotient by the natural Z2 action on S3.

20.4. Double fibration of SO(3) and integral geometry on S2

In this section, we present a self-contained account of an integral-geometric identity, used in the proof of Pu’s inequality (17.1.6) in Sec-tion 20.5, as well as in our argument in Lemma The identity has itsorigin in results of P. Funk [Fu16] determining a symmetric functionon the two-sphere from its great circle integrals; see [Hel99, Proposi-tion 2.2, p. 59], as well as Preface therein.

The sphere S2 is the homogeneous space of the Lie group SO(3),cf. (20.3.2), so that S2 = SO(3)/SO(2), cf. [Ar83, p. 82]. Denoteby SO(2)σ the fiber over (stabilizer of) a typical point σ ∈ S2. Theprojection

p : SO(3) → S2 (20.4.1)

is a Riemannian submersion for the standard metric of constant sec-tional curvature 1

4on SO(3). The total space SO(3) admits another

Riemannian submersion, which we denote

q : SO(3) → RP2, (20.4.2)

whose typical fiber ν is an orbit of the geodesic flow on SO(3) when thelatter is viewed as the unit tangent bundle of S2, cf. Remark 20.3.1.

Here RP2 denotes the double cover of RP2, homeomorphic to the 2-sphere, cf. Definition 17.20.2. A fiber of fibration q is the collection of


SO(2)

##GGG

GGGG

GGG

(RP2, dν

)

SO(3)

q99ttttttttt

p

&&LLL

LLLL

LLL

ν

::uuuuuuuuuu

great circle// (S2, dσ)

Figure 20.4.1. Integral geometry on S2, cf. (20.4.1) and (20.4.2)

unit vectors tangent to a given directed closed geodesic (great circle)on S2. Thus, each orbit ν projects under p to a great circle on the

sphere. We think of the base space RP2 of q as the configuration spaceof oriented great circles on the sphere, with measure dν. While fibra-tion q may seem more “mysterious” than fibration p, the two are actu-ally equivalent from the quaternionic point of view, cf. Remark 20.3.2.

The diagram of Figure 20.4.1 illustrates the maps defined so far.Denote by Eg(ν) the energy, and by Lg(ν) the length, of a curve ν

with respect to a (possibly singular) metric g = f 2g0.

Theorem 20.4.1. We have the following integral-geometric iden-tity:

area(g) =1

2π

∫

˜RP2

Eg(ν)dν.

Proof. We apply Fubini’s theorem [Ru87, p. 164] twice, to both qand p, to obtain

∫

˜RP2

Eg(ν)dν =

∫

˜RP2

dν

(∫

ν

f 2 ◦ p ◦ ν(t)dt)

=

∫

SO(3)

f 2 ◦ p

=

∫

S2

f 2 dσ

(∫

SO(2)σ

1

)

= 2π area(g),

(20.4.3)

completing the proof of the theorem. �

Proposition 20.4.2. Let f be a square integrable function on S2,which is positive and continuous except possibly for a finite number of

20.6. A TABLE OF OPTIMAL SYSTOLIC RATIOS OF SURFACES 221

points where f either vanishes or has a singularity of type 1√r, Then

there is a great circle ν such that the following inequality is satisfied:(∫

ν

f(ν(t))dt

)2

≤ π

∫

S2

f 2(σ)dσ, (20.4.4)

where t is the arclength parameter, and dσ is the Riemannian measureof the metric g0 of the standard unit sphere. In other words, there is agreat circle of length L in the metric f 2g0, so that L2 ≤ πA, where A isthe Riemannian surface area of the metric f 2g0. In the boundary caseof equality in (20.4.4), the function f must be constant.

Proof. The proof is an averaging argument and shows that theaverage length of great circles is short. Comparing length and energyyields the inequality(∫

˜RP2 Lg(ν)2)

2π≤∫

˜RP2

Eg(ν)dν. (20.4.5)

The formula now follows from Theorem 20.4.1, since area(RP 2) = 4π.In the boundary case of equality in (20.4.4), one must have equality

also in inequality (20.4.5) relating length and energy. It follows that theconformal factor f must be constant along every great circle. Hence fis constant everywhere on S2. �

20.5. Proof of Pu’s inequality

A metric g on RP2 lifts to a centrally symmetric metric g on S2.Applying Proposition 20.4.2 to g, we obtain a great circle of g-length Lsatisfying L2 ≤ πA, where A is the Riemannian surface area of g. Thus

we have(L2

)2 ≤ π2

(A2

), where sys1(g) ≤ L

2while area(g) = A

2, proving

inequality (17.1.6). See [Iv02] for an alternative proof.

20.6. A table of optimal systolic ratios of surfaces

Denote by SR(M) the supremum of the systolic ratios,

SR(M) = supg

sys1(g)2

area(g),

ranging over all metrics g on a surface M . The known values of theoptimal systolic ratio are tabulated in Figure 20.6.1. It is interesting tonote that the optimal ratio for the Klein bottle RP2#RP2 is achievedby a singular metric, described in the references listed in the table.


SR(M) numerical value where to find it

M = RP2 =π

2[Pu52] ≈ 1.5707 Section 20.5

infinite π1(M) <4

3[Gro83] < 1.3333 . . . (17.1.7)

M = T2 = 2√3(Loewner) ≈ 1.1547 (17.2.1)

M =RP2#RP2

=π

23/2≈ 1.1107 [Bav86, Bav06,

Sak88]

M of genus 2 > 13

(√2 + 1

)> 0.8047 ?

M of genus 3 ≥ 87√3

> 0.6598 [Cal96]

Figure 20.6.1. Values for optimal systolic ratio SR of surface M

CHAPTER 21

A primer on surfaces

In this Chapter, we collect some classical facts on Riemann surfaces.More specifically, we deal with hyperelliptic surfaces, real surfaces, andKatok’s optimal bound for the entropy of a surface.

21.1. Hyperelliptic involution

Let M be an orientable closed Riemann surface which is not asphere. By a Riemann surface, we mean a surface equipped with afixed conformal structure, cf. Definition 17.15.4, while all maps areangle-preserving.

Furthermore, we will assume that the genus is at least 2.

Definition 21.1.1. A hyperelliptic involution of a Riemann sur-face M of genus p is a holomorphic (conformal) map, J : M → M ,satisfying J2 = 1, with 2p + 2 fixed points.

Definition 21.1.2. A surface M admitting a hyperelliptic involu-tion will be called a hyperelliptic surface.

Remark 21.1.3. The involution J can be identified with the non-trivial element in the center of the (finite) automorphism group of M(cf. [FK92, p. 108]) when it exists, and then such a J is unique,cf. [Mi95, p.204] (recall that the genus is at least 2).

It is known that the quotient of M by the involution J produces aconformal branched 2-fold covering

Q :M → S2 (21.1.1)

of the sphere S2.

Definition 21.1.4. The 2p + 2 fixed points of J are called Weier-strass points. Their images in S2 under the ramified double cover Q offormula (21.1.1) will be referred to as ramification points.

A notion of a Weierstrass point exists on any Riemann surface, butwill only be used in the present text in the hyperelliptic case.

223

224 21. A PRIMER ON SURFACES

Example 21.1.5. In the case p = 2, topologically the situationcan be described as follows. A simple way of representing the figure 8contour in the (x, y) plane is by the reducible curve

(((x− 1)2 + y2)− 1)(((x+ 1)2 + y2)− 1) = 0 (21.1.2)

(or, alternatively, by the lemniscate r2 = cos 2θ in polar coordinates,i.e. the locus of the equation (x2 + y2)2 = x2 − y2).

Now think of the figure 8 curve of (21.1.2) as a subset of R3. Theboundary of its tubular neighborhood in R3 is a genus 2 surface. Rota-tion by π around the x-axis has six fixed points on the surface, namely,a pair of fixed points near each of the points −2, 0, and +2 on the x-axis. The quotient by the rotation can be seen to be homeomorphic tothe sphere.

A similar example can be repackaged in a metrically more preciseway as follows.

Example 21.1.6. We start with a round metric on RP2. Nowattach a small handle. The orientable double cover M of the resultingsurface can be thought of as the unit sphere in R3, with two littlehandles attached at north and south poles, i.e. at the two points wherethe sphere meets the z-axis. Then one can think of the hyperellipticinvolution J as the rotation of M by π around the z-axis. The sixfixed points are the six points of intersection of M with the z-axis.Furthermore, there is an orientation reversing involution τ onM , givenby the restriction toM of the antipodal map in R3. The composition τ◦J is the reflection fixing the xy-plane, in view of the following matrixidentity:

−1 0 00 −1 00 0 −1

−1 0 00 −1 00 0 1

=

1 0 00 1 00 0 −1

. (21.1.3)

Meanwhile, the induced orientation reversing involution τ0 on S2 canjust as well be thought of as the reflection in the xy-plane. This isbecause, at the level of the 2-sphere, it is “the same thing as” thecomposition τ ◦ J . Thus the fixed circle of τ0 is precisely the equator,cf. formula (21.3.3). Then one gets a quotient metric on S2 which isroughly that of the western hemisphere, with the boundary longitudefolded in two, The metric has little bulges along the z-axis at northand south poles, which are leftovers of the small handle.

21.3. OVALLESS SURFACES 225

21.2. Hyperelliptic surfaces

For a treatment of hyperelliptic surfaces, see [Mi95, p. 60-61].By [Mi95, Proposition 4.11, p. 92], the affine part of a hyperellip-tic surface M is defined by a suitable equation of the form

w2 = f(z) (21.2.1)

in C2, where f is a polynomial. On such an affine part, the map J isgiven by J(z, w) = (z,−w), while the hyperelliptic quotient map Q :M → S2 is represented by the projection onto the z-coordinate in C2.

A slight technical problem here is that the map

M → CP2, (21.2.2)

whose image is the compactification of the solution set of (21.2.1), isnot an embedding. Indeed, there is only one point at infinity, givenin homogeneous coordinates by [0 : w : 0]. This point is a singularity.A way of desingularizing it using an explicit change of coordinates atinfinity is presented in [Mi95, p. 60-61]. The resulting smooth surfaceis unique [DaS98, Theorem, p. 100].

Remark 21.2.1. To explain what happens “geometrically”, notethat there are two points on our affine surface “above infinity”. Thismeans that for a large circle |z| = r, there are two circles above it satis-fying equation (21.2.1) where f has even degree 2p+2 (for a Weierstrasspoint we would only have one circle). To see this, consider z = reia. Asthe argument a varies from 0 to 2π, the argument of f(z) will changeby (2p + 2)2π. Thus, if (reia, w(a)) represents a continuous curve onour surface, then the argument of w changes by (2p+2)π, and hence weend up where we started, and not at −w (as would be the case were thepolynomial of odd degree). Thus there are two circles on the surfaceover the circle |z| = r. We conclude that to obtain a smooth compactsurface, we will need to add two points at infinity, cf. discussion around[FK92, formula (7.4.1), p. 102].

Thus, the affine part of M , defined by equation (21.2.1), is a Rie-mann surface with a pair of punctures p1 and p2. A neighborhood ofeach pi is conformally equivalent to a punctured disk. By replacingeach punctured disk by a full one, we obtain the desired compact Rie-mann surface M . The point at infinity [0 : w : 0] ∈ CP2 is the imageof both pi under the map (21.2.2).

21.3. Ovalless surfaces

Denote by M ι the fixed point set of an involution ι of a Riemannsurface M . Let M be a hyperelliptic surface of even genus p. Let J :


M → M be the hyperelliptic involution, cf. Definition 21.1.1. Let τ :M →M be a fixed point free, antiholomorphic involution.

Lemma 21.3.1. The involution τ commutes with J and descendsto S2. The induced involution τ0 : S

2 → S2 is an inversion in a circleC0 = Q(M τ◦J ). The set of ramification points is invariant under theaction of τ0 on S2.

Proof. By the uniqueness of J , cf. Remark 21.1.3, we have thecommutation relation

τ ◦ J = J ◦τ, (21.3.1)

cf. relation (21.1.3). Therefore τ descends to an involution τ0 on thesphere. There are two possibilities, namely, τ is conjugate, in the groupof fractional linear transformations, either to the map z 7→ z, or to themap z 7→ −1

z. In the latter case, τ is conjugate to the antipodal map

of S2.In the case of even genus, there is an odd number of Weierstrass

points in a hemisphere. Hence the inverse image of a great circle is aconnected loop. Thus we get an action of Z2 × Z2 on a loop, resultingin a contradiction.

In more detail, the set of the 2p + 2 ramification points on S2 iscentrally symmetric. Since there is an odd number, p + 1, of ramifi-cation points in a hemisphere, a generic great circle A ⊆ S2 has theproperty that its inverse image Q−1(A) ⊆ M is connected. Thus bothinvolutions τ and J , as well as τ ◦ J , act fixed point freely on theloop Q−1(A) ⊆M , which is impossible. Therefore τ0 must fix a point.It follows that τ0 is an inversion in a circle. �

Suppose a hyperelliptic Riemann surfaceM admits an antiholomor-phic involution τ . In the literature, the components of the fixed pointsetM τ of τ are sometimes referred to as “ovals”. When τ is fixed pointfree, we introduce the following terminology.

Definition 21.3.2. A hyperelliptic surface (M,J) of even positivegenus p > 0 is called ovalless real if one of the following equivalentconditions is satisfied:

(1) M admits an imaginary reflection, i.e. a fixed point free, an-tiholomorphic involution τ ;

(2) the affine part of M is the locus in C2 of the equation

w2 = −P (z), (21.3.2)

where P is a monic polynomial, of degree 2p + 2, with realcoefficients, no real roots, and with distinct roots.

21.4. KATOK’S ENTROPY INEQUALITY 227

Lemma 21.3.3. The two ovalless reality conditions of Definition 21.3.2are equivalent.

Proof. A related result appears in [GroH81, p. 170, Proposi-tion 6.1(2)]. To prove the implication (2) =⇒ (1), note that complexconjugation leaves the equation invariant, and therefore it also leavesinvariant the locus of (21.3.2). A fixed point must be real, but P is pos-itive hence (21.3.2) has no real solutions. There is no real solution atinfinity, either, as there are two points at infinity which are not Weier-strass points, since P is of even degree, as discussed in Remark 21.2.1.The desired imaginary reflection τ switches the two points at infinity,and, on the affine part of the Riemann surface, coincides with complexconjugation (z, w) 7→ (z, w) in C2.

To prove the implication (1) =⇒ (2), note that by Lemma 21.3.1,the induced involution τ0 on S

2 =M/ J may be thought of as complexconjugation, by choosing the fixed circle of τ0 to be the circle

R ∪ {∞} ⊆ C ∪ {∞} = S2. (21.3.3)

Since the surface is hyperelliptic, it is the smooth completion of thelocus in C2 of some equation of the form (21.3.2), cf. (21.2.1). Here Pis of degree 2p+2 with distinct roots, but otherwise to be determined.The set of roots of P is the set of (the z-coordinates of) the Weierstrasspoints. Hence the set of roots must be invariant under τ0. Thus theroots of the polynomial either come in conjugate pairs, or else are real.Therefore P has real coefficients. Furthermore, the leading cofficientof P may be absorbed into the w-coordinate by extracting a squareroot. Here we may have to rotate w by i, but at any rate the coefficientsof P remain real, and thus P can be assumed monic.

If P had a real root, there would be a ramification point fixed by τ0.But then the corresponding Weierstrass point must be fixed by τ , aswell! This contradicts the fixed point freeness of τ . Thus all roots of Pmust come in conjugate pairs. �

21.4. Katok’s entropy inequality

Let (M, g) be a closed surface with a Riemannian metric. De-

note by (M, g) the universal Riemannian cover of (M, g). Choose apoint x0 ∈ M .

Definition 21.4.1. The volume entropy (or asymptotic volume) h(M, g)of a surface (M, g) is defined by setting

h(M, g) = limR→+∞

log (volg B(x0, R))

R, (21.4.1)


where volg B(x0, R) is the volume (area) of the ball of radius R centered

at x0 ∈ M .

Since M is compact, the limit in (21.4.1) exists, and does not de-

pend on the point x0 ∈ M [Ma79]. This asymptotic invariant describesthe exponential growth rate of the volume in the universal cover.

Definition 21.4.2. The minimal volume entropy, MinEnt, ofM isthe infimum of the volume entropy of metrics of unit volume on M , orequivalently

MinEnt(M) = infgh(M, g) vol(M, g)

12 (21.4.2)

where g runs over the space of all metrics on M . For an n-dimensionalmanifold in place of M , one defines MinEnt similarly, by replacing theexponent of vol by 1

n.

The classical result of A. Katok [Kato83] states that every metric gon a closed surfaceM with negative Euler characteristic χ(M) satisfiesthe optimal inequality

h(g)2 ≥ 2π|χ(M)|area(g)

. (21.4.3)

Inequality (21.4.3) also holds for hom ent(g) [Kato83], as well as thetopological entropy, since the volume entropy bounds from below thetopological entropy (see [Ma79]). We recall the following well-knownfact, cf. [KatH95, Proposition 9.6.6, p. 374].

Lemma 21.4.3. Let (M, g) be a closed Riemannian manifold. Then,

h(M, g) = limT→+∞

log(P ′(T ))

T(21.4.4)

where P ′(T ) is the number of homotopy classes of loops based at somefixed point x0 which can be represented by loops of length at most T .

Proof. Let x0 ∈M and choose a lift x0 ∈ M . The group

Γ := π1(M,x0)

acts on M by isometries. The orbit of x0 under Γ is denoted Γ.x0.Consider a fundamental domain ∆ for the action of Γ, containing x0.Denote by D the diameter of ∆. The cardinal of Γ.x0 ∩ B(x0, R) isbounded from above by the number of translated fundamental do-mains γ.∆, where γ ∈ Γ, contained in B(x0, R+D). It is also bounded

21.4. KATOK’S ENTROPY INEQUALITY 229

from below by the number of translated fundamental domains γ.∆contained in B(x0, R). Therefore, we have

vol(B(x0, R))

vol(M, g)≤ card (Γ.x0 ∩B(x0, R)) ≤

vol(B(x0, R +D))

vol(M, g).

(21.4.5)Take the log of these terms and divide by R. The lower bound becomes

1

Rlog

(vol(B(R))

vol(g)

)=

=1

Rlog (vol(B(R)))− 1

Rlog (vol(g)) ,

(21.4.6)

and the upper bound becomes

1

Rlog

(vol(B(R +D))

vol(g)

)=

=R +D

R

1

R +Dlog(vol(B(R +D)))− 1

Rlog(vol(g)).

(21.4.7)

Hence both bounds tend to h(g) when R goes to infinity. Therefore,

h(g) = limR→+∞

1

Rlog (card(Γ.x0 ∩B(x0, R))) . (21.4.8)

This yields the result since P ′(R) = card(Γ.x0 ∩ B(x0, R)). �

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