Differential geometry and topology

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. 620/00-10.0 and 84/03).

Abstract. We proceed from the classical topics in differential ge- ometry such as the geodesic equation and Gauss’s theorema egregium, to systolic geometry in its arithmetic, ergodic, and topological manifestations. Contents

Chapter 1. Introduction 5 Chapter 2. Linear algebra, lattices, curves 7 2.1. Index notation 7 2.2. Finding an eigenvector of a symmetric matrix 10 2.3. Hessian, minima, maxima, saddle points 13 2.4. Successive minima of a lattice 13 2.5. Hermite and Berg´e-Martinet constants 16 2.6. Curves and their curvature 18 2.7. Great circles and Clairaut’s relation 21 Chapter 3. Local geometry of surfaces 27 3.1. First fundamental form 27 3.2. Length and area 30 3.3. Christoffel symbols 31 3.4. What is a geodesic on a surface? 34 3.5. Geodesics on a surface of revolution 36 3.6. Weingarten map 37 3.7. Gaussian curvature 41 3.8. Calculus of variations and the geodesic equation 43 3.9. Principal curvatures 45 3.10. Minimal surfaces 46 Chapter 4. The theorema egregium of Gauss 49 4.1. Intrinsic vs extrinsic properties 49 4.2. Preliminaries to the theorema egregium 50 4.3. The theorema egregium of Gauss 53 4.4. The Laplacian formula for Gaussian curvature 54 Chapter 5. Global geometry of surfaces 57 5.1. Metric preliminaries 57 5.2. Geodesic equation and closed geodesics 60 5.3. Surfaces of constant curvature 62 5.4. Flat surfaces 64 5.5. Hyperbolic surfaces 65 3 4 CONTENTS

5.6. Topological preliminaries 66 Chapter 6. Inequalities of Loewner and Pu 69 6.1. Definition of systole 69 6.2. Isoperimetric inequality and Pu’s inequality 69 6.3. Hermite and Berg´e-Martinet constants 71 6.4. The Loewner inequality 73 Chapter 7. Systolic applications of integral geometry 75 7.1. An integral-geometric identity 75 7.2. Two proofs of the Loewner inequality 76 7.3. Hopf fibration and the Hamilton quaternions 78 7.4. Double fibration of SO(3) and integral geometry on S2 79 7.5. Proof of Pu’s inequality 81 7.6. A table of optimal systolic ratios of surfaces 81 Chapter 8. A primer on surfaces 83 8.1. Hyperelliptic involution 83 8.2. Hyperelliptic surfaces 84 8.3. Ovalless surfaces 85 8.4. Katok’s entropy inequality 87 Bibliography 91 CHAPTER 1

Introduction

These notes are based on a course taught at Bar Ilan University. After dealing with classical geometric preliminaries including the the- orema egregium of Gauss, we present new geometric inequalities on Riemann surfaces, as well as their higher dimensional generalisations.

5

CHAPTER 2

Linear algebra, lattices, curves

2.1. Index notation Let Rn denote the Euclidean n-space. Its vectors will be denoted v, w Rn. Let B be an n by n matrix. There are two ways of viewing such a ∈ matrix, either as a linear map or as a bilinear form (cf. Remarks 2.1.1 and 2.1.3). Developing suitable notation to capture this distinction helps simplify differential-geometric formulas down to readable size, and ultimately to motivate the crucial distinction between a vector and a covector. Remark 2.1.1 (Matrix as a bilinear form B(v, w)). Consider a bilinear form B(v, w) : Rn Rn R, (2.1.1) × → sending the pair of vectors (v, w) to the real number vtBw. Here vt is the transpose of v. We write

B = (bij)i=1,...,n; j=1,...,n so that bij is an entry while (bij) denotes the matrix. For example, in the 2 by 2 case, v1 w1 b b v = , w = , B = 11 12 . v2 w2 b b      21 22 Then b b w1 b w1 + b w2 Bw = 11 12 = 11 12 . b b w2 b w1 + b w2  21 22    21 22  Now the transpose vt = (v1 v2) is a row vector, so B(v, w) = vtBw b w1 + b w2 = (v1 v2) 11 12 b w1 + b w2  21 22  1 1 1 2 2 1 2 2 = b11v w + b12v w + b21v w + b22v w ,

t 2 2 i j and therefore v Bw = i=1 j=1 bijv w . We would like to simplify this notation by deleting the summation symbols “Σ”, as follows. P P 7 8 2. LINEAR ALGEBRA, LATTICES, CURVES

Definition 2.1.2 (A notation called “Einstein summation conven- tion”). The rule is that whenever a product contains a symbol with a lower index and another symbol with the same upper index, take summation over this repeated index. Using this notation, the bilinear form (2.1.1) defined by the ma- trix B can be written as follows: i j B(v, w) = bijv w , with implied summation over both indices. To write down the asso- i ciated quadratic form Q(v) = B(v, v), let v = v ei where ei is the standard basis of Rn. To compute Q(v), we must introduce{ an} extra index j : i j i j i j Q(v) = B(v, v) = B(v ei, v ej) = B(ei, ej)v v = bijv v . (2.1.2) Recall that the polarisation formula allows one to reconstruct the bi- linear form from the quatratic form, at least if the characteristic is not 2: 1 B(v, w) = (Q(v + w) Q(v w)). 4 − − Remark 2.1.3 (Matrix as a linear map B : Rn Rn, v Bv). In order to distinguish this case from the case of the→bilinear form,7→ we will raise the first index : write B as i B = (b j)i=1,...,n; j=1,...,n where it is important to s t a g g e r the indices, meaning that we do not put j under i as in i bj, but, rather, leave a blank space (in the place were j used to be), as in i b j.

j i Let v = (v )j=1,...,n and w = (w )i=1,...,n. Then the equation w = Bv i i j can be written as a system of n scalar equations, w = b jv for i = 1, . . . , n using the Einstein summation convention (here the repeated index is j).

1 Remark 2.1.4 (Trace). The formula for the trace T r(B) = b 1 + b2 + + bn in Einstein notation becomes 2 · · · n i T r(B) = b i (here the repeated index is i). 2.1. INDEX NOTATION 9

Remark 2.1.5 (Symmetrisation and skew-symmetrisation). Let B = (bij). Its symmetric part S is by definition

1 t 1 S = (B + B ) = (bij + bji) i=1,...,n , 2 2 j=1,...,n   while the skew-symmetric part A is

1 t 1 A = (B B ) = (bij bji) i=1,...,n . 2 − 2 − j=1,...,n   Note that B = S + A. Another useful notation is that of symmetrisation 1 b ij = (bij + bji) (2.1.3) { } 2 and antisymmetrisation 1 b = (b b ). (2.1.4) [ij] 2 ij − ji

Lemma 2.1.6. A matrix B = (bij) is symmetric if and only if for all indices i and j one has b[ij] = 0. Proof. We have b = 1 (b b ) = 0 since symmetry of B [ij] 2 ij − ji means bij = bji.  Remark 2.1.7 (Matrix multiplication in index notation). The usual way to define matrix multiplication is as follows. A triple of matri- ces A = (aij), B = (bij), and C = (cij) satisfy the product relation C = AB if, introducing an additional dummy index k (cf. formula (2.1.2)), we have the relation cij = aikbkj. Xk Example 2.1.8 (Skew-symmetrisation of matrix product). By com- mutativity of multiplication of real numbers, aikbkj = bkjaik. Then the coefficients c[ij] of the skew-symmetrisation of C = AB satisfy

c[ij] = bk[jai]k. Xk Here by definition 1 b a = (b a b a ). k[j i]k 2 kj ik − ki jk This notational device will be particularly useful in writing down the theorema egregium. Given below are a few examples: See section 3.3, where we will use formulas of type • m m m gmjΓik + gmiΓjk = 2gm jΓi k; { } 10 2. LINEAR ALGEBRA, LATTICES, CURVES

section 3.7 for Li Lk ; • [j l] section 4 for Γk Γn . • i[j l]m The index notation we have described reflects the fact that the nat- ural products of matrices are the ones which correspond to composition i i i of maps. Thus, if A = (a j), B = (b j), and C = (c j) then the product i i k relation C = AB simplifies to the relation c j = a kb j.

Remark 2.1.9 (The inverse matrix). Let B = (bij). The in- 1 1 ij verse B− is sometimes written as B− = (b ) (here both indices have been raised).

1 ik i Then the equation B− B = I becomes b bkj = δ j (in Einstein notation with repeated index k), where the expression 1 if i = j δi = j 0 if i = j  6 is referred to as the Kronecker delta function.

i Example 2.1.10. The identity endomorphism I = (δ j) by defi- i nition satisfies AI = A = IA for all endomorphisms A = (a j), or equivalently i j i i j a jδ k = a k = δ ja k, using the Enstein summation convention.

i i Definition 2.1.11. Given a pair of vectors v = v ei and w = w ei in R3, their vector product is a vector v w R3 satisfying one of the following two equivalent conditions: × ∈

e1 e2 e3 (1) we have v w = det v1 v2 v3 , in other words, × w1 w2 w3 v w = (v2w3 v3w2)e (v1w3 v3w1)e + (v1w2 v2w1)e × − 1 − − 2 − 3 = 2 v[2w3]e v[1w3]e + v[1w2]e . 1 − 2 3 (2) the vector v w is perpendicular
to both v and w, of length equal to the×area of the parallelogram spanned by the two vectors, and furthermore satisfying the right hand rule, mean- ing that the 3 by 3 matrix formed by the three vectors v, w, and v w has positive determinant. × 2.2. Finding an eigenvector of a symmetric matrix In this section we prove the existence of a real eigenvector (and hence, a real eigenvalue) for a real symmetric matrix (in general, a real 2.2. FINDING AN EIGENVECTOR OF A SYMMETRIC MATRIX 11 matrix may not have such an eigenvector). What does this have to do with surfaces? The answer is that the various curvatures of a surface are defined in terms of the eigenvalues of a certain symmetric matrix, or more precisely selfadjoint operator (see sections 3.6 and 3.9). Let 1 0 . I = ..   0 1   be the (n, n) identity matrix. Thus I = (δij) i=1,...,n where δij is the j=1,...,n Kronecker delta. Let B be an (n, n)-matrix. Definition 2.2.1. A real number λ is called an eigenvalue of B if det(B λI) = 0. − Theorem 2.2.2. If λ R is an eigenvalue of B, then there is a vector v Rn, v = 0, such∈that Bv = λv. ∈ 6 The proof is standard. Such a vector is called an eigenvector belong- ing to λ. Recall the formula for the Euclidean inner product: v, w = 1 1 n n n i i h it v w + +v w = i=1 v w . We may view it as follows : v, w = v w. Recall ·the· · following property of the transpose: (AB)t = Bh tAt.i P Lemma 2.2.3. A real matrix B is symmetric if and only if for all v, w Rn, one has ∈ Bv, w = v, Bw . h i h i Proof. We have Bv, w = (Bv)tw = vtBtw = v, Btw = v, Bw . h i h i h i

Theorem 2.2.4. Every real symmetric matrix possesses a real eigen- vector. We will give two proofs of this important theorem. The first proof has the advantage of being more geometric, as well as more concrete in the choice of the vector in question. The second proof is simpler, but passes via complexification. First proof. Let S Rn be the unit sphere : ⊂ S = v Rn v = 1 . { ∈ | k k } Given a symmetric matrix B, define a function f : S R as the restriction to S of the function, also denoted f, given by → f(v) = v, Bv . h i 12 2. LINEAR ALGEBRA, LATTICES, CURVES

n Let v be a maximum of f restricted to S. Let V ⊥ R be the orthog- 0 0 ⊂ onal complement of the line spanned by v . Let w V ⊥. Consider the 0 ∈ 0 curve v0 +tw, t 0 (see also a different choice of curve in Remark 2.2.5 d≥ below). Then f(v + tw) = 0 since v is a maximum and w is dt 0 0 t=0 tangent to the sphere.

Now d d f(v + tw) = v + tw, B(v + tw) dt 0 dt h 0 0 i 0 0

d = ( v , Bv + t v , Bw + t w, Bv + t2 w, Bv ) dt h 0 i h 0 i h 0i h i 0

= v0 , Bw + w, Bv0 h i h i = Bw, v + Bv , w h 0i h 0 i = Btv , w + Bv , w h 0 i h 0 i = (Bt + B)v , w h 0 i = 2 Bv , w by symmetry of B. h 0 i Thus Bv , w = 0 for all w V ⊥. Hence Bv is proportional to v h 0 i ∈ 0 0 0 and so v0 is an eigenvector of B. 

Remark 2.2.5. Our calculation used the curve v0 +tw which, while tangent to S at v0 (see section 2.7), does not lie on S. If one prefers, one can use instead the curve (cos t)v0 + (sin t)w lying on S. Then d t (cos t)v0+(sin t)w, B((cos t)v0+(sin t)w) = = (B +B)v0, w dt h i · · · h i t=0 and one argues by symmetry as before.

Second proof. Let B be an n n real symmetric matrix, where n × n n ≥ 1. As such, it defines a linear map BR : R R . Meanwhile, viewed as a complex matrix, the matrix B defines a→complex linear map

n n BC : C C . → The characteristic polynomial of BC is a polynomial of positive de- gree n > 0 and therefore has a root λ by the fundamental theorem of algebra. Let v Cn be an associated eigenvector. Let , be the stan- dard Hermitian∈inner product in Cn. Then Bv, v =h v,iBv since B h i h i is real symmetric. Hence λv, v = v, λv = λ v, v , and therefore λ is real. h i h i h i  2.4. SUCCESSIVE MINIMA OF A LATTICE 13

Definition 2.2.6. Let V be a real inner product space. An endo- morphism B : V V is selfadjoint if for all v, w V , one has Bv, w = v, Bw . → ∈ h i h i Corollary 2.2.7. Every selfadjoint endomorphism of an inner product space admits a real eigenvector. Proof. The selfadjointness was the relevant property in the proof of Theorem 2.2.4.  2.3. Hessian, minima, maxima, saddle points

1 n Given a smooth function f(u , . . . , u ) of n variables, denote by Hf the Hessian matrix of f, i.e. the n n matrix of second partial deriva- ∂2f × tives fij = ∂ui∂uj of f. The familiar property of the equality of mixed partials can be written in terms of the antisymmetrisation notation defined above (2.1.4) as follows:

f[ij] = 0. (2.3.1) ∂f A critical point p of f is a point satisfying f(p) = 0, i.e. ∂ui (p) = 0 for all i = 1, . . . , n. ∇ Example 2.3.1. Let n = 2. Then the sign of the determinant ∂2f ∂2f ∂2f 2 det(H ) = f ∂u1∂u1 ∂u2∂u2 − ∂u1∂u2   at a critical point has geometric significance. Namely, if

det(Hf (p)) > 0, then p is a maximum or a minimum. If det(Hf (p)) < 0, then p is a saddle point. Moreover, the value of Hf (p) also has geometric signif- icance. Namely, it is precisely the Gaussian curvature of the surface which is the graph of f in R3, cf. Definition 3.7.1. 2.4. Successive minima of a lattice Let b > 0. By a lattice L in Euclidean space Rb, we mean a discrete subgroup isomorphic to Zb, i.e. the integer span of a linearly indepen- dent set of b vectors. Definition 2.4.1. An orbit of a point x Rb under L is the subset of Rb given by the collection of elements ∈ g + x g L . { | ∈ } The quotient Rb/L (which in this case can be understood at the group-theoretic level) is called a b-torus. 14 2. LINEAR ALGEBRA, LATTICES, CURVES

Definition 2.4.2. A fundamental domain for the torus Rb/L is a closed set F Rb satisfying the following three conditions: ⊂ every orbit meets F in at least one point; • every orbit meets the interior of F in at most one point; • the boundary ∂F is of zero Lebesgue measure. • Definition 2.4.3. The total volume of the b-torus Rb/L is by def- inition the b-volume of a fundamental domain. It is shown in advanced calculus that the total volume thus defined is independent of the choice of a fundamental domain.

Example 2.4.4. Let b = 1. Every lattice in R is of the form Lα = αZ R, for some real α = 0. It is spanned by the vector αe1 (or αe1). The⊂ quotient in this 6 case is a circle, see Theorem 2.4.5. A −funda- mental domain for the quotient may be chosen to be the interval [0, α]. The 1 volume, i.e. the length, of the quotient circle is therefore pre- cisely α−. More background needed from complex analysis and ez.

Theorem 2.4.5. The quotient group R/Lα is isomorphic to the circle S1 C. ⊂ Proof. Consider the map φˆ : R C defined by → i2πx φˆ(x) = e α . This map is a homomorphism since φˆ(x+y) = φˆ(x)φˆ(y). Furthermore, we have φˆ(x + αm) = φˆ(x) for all m Z. Hence φˆ descends to a map ∈ 1 φ : R/Lα C, which is injective. Its image is the unit circle S C, which is a→group under multiplication. ⊂ 

2 Example 2.4.6. Let b = 2. Every lattice in R is of the form Lα = 2 SpanZ(v, w) R , where v, w is a linearly independent set. For ⊂ { } example, let α and β be nonzero reals. Set Lα,β = SpanZ(αe1, βe2) R2. This lattice admits an orthogonal basis, namely αe , βe . ⊂ { 1 2} More generally, let B be a finite-dimensional Banach space, i.e. a vector space together with a norm . k k Lemma 2.4.7. Banach space (B, ) is isometric to Euclidean space if either one of the following two equivalentk k conditions is satisfied: (1) the unit ball of the norm is an ellipsoid; (2) the norm is defined by a krekal inner product , by the for- h i mula v = v, v . k k h i p 2.4. SUCCESSIVE MINIMA OF A LATTICE 15

Proof. Choose a basis (vi) of B consisting of the half-axes of the b i i ellipsoid, and define a map f : B R by f(a vi) = a ei. Clearly, the map is an isometry. Therefore→the norm is defined by an inner product.  Let L (B, ) be a lattice of maximal rank rank(L) = dim(B). ⊂ k k Definition 2.4.8. For each k = 1, 2, . . . , rank(L), define the k-th successive minimum of the lattice L by

lin. indep. v1, . . . , vk L λk(L, ) = inf λ R ∃ ∈ . (2.4.1) k k ∈ with vi λ  k k ≤ 

Example 2.4.9. The first successive minimum, λ1(L, ) is the least length of a nonzero vector in L. k k

We illustrate the geometric meaning of λ1 in terms of the circle of Theorem 2.4.5. Theorem 2.4.10. Consider a lattice L R. Then the circle R/L satisfies ⊂ length(R/L) = λ1(L). Proof. The proof is immediate from the discussion in Example 2.4.4 above.  Example 2.4.11. Assume rank(L) 2. Given a pair of vec- tors S = v, w in L, define the length≥ S of S by setting S = { } | | | | max( v , w ). Then the second successive minimum, λ2(L, ) is the least klengthk k ofk a pair of non-proportional vectors in L: k k

λ2(L) = inf S , S | | where S runs over all linearly independent (i.e. non-proportional) pairs v, w L. { } ⊂ The volume of the torus Rb/L is also called the covolume of the lattice L. Is by definition the volume of a fundamental domain for L, e.g. a parallelepiped spanned by a Z-basis for L. The volume can be calculated as the square root of the determinant of the Gram matrix of such a basis, cf. (2.4.2). b Definition 2.4.12. Given a finite set S = vi in R , we define its Gram matrix as the matrix of inner products { } Gram(S) = ( v , v ) . h i ji i=1,...,n; j=1,...,n Then the parallelogram P spanned by the vectors v satisfies { i} vol(P ) = det(Gram(S)). (2.4.2) p 16 2. LINEAR ALGEBRA, LATTICES, CURVES

Example 2.4.13 (Gaussian integers). The standard lattice Zb Rb b b b ⊂b b satisfies λ1(Z ) = . . . = λb(Z ) = 1. Note that the torus T = R /Z satisfies vol(Tb) = 1 as it has the unit cube as a fundamental domain. In dimension 2, the resulting lattice in C = R2 is called the Gaussian integers. It contains 4 elements of least length, namely the fourth roots of unity.

Example 2.4.14 (Eisenstein integers). Consider the lattice Lζ 2 2πi ⊂ R = C spanned by 1 C and the sixth root of unity ζ = e 6 C. ∈ ∈ The resulting lattice is called the Eisenstein integers. One has Lζ 2 2 satisfies λ1(Lζ ) = λ2(Lζ ) = 1. Meanwhile, the torus T = R /Lζ T2 √3 satisfies area( ) = 2 . It contains 6 elements of least length, namely the sixth roots of unity.

2.5. Hermite and Berg´e-Martinet constants

The Hermite constant γb is defined in one of the following two equiv- alent ways:

(1) γb is the square of the maximal first successive minimum, among all lattices of unit covolume; (2) γb is defined by the formula

λ1(L) Rb √γb = sup 1 L ( , ) , (2.5.1) ( vol(Rb/L) b ⊆ k k )

Rb where the supremum is extended over all lattices L in with a Euclidean norm . k k A lattice realizing the supremum may be thought of as the one real- Rb 1 izing the densest packing in when we place the balls of radius 2 λ1(L) at the points of L. We will discuss the case b = 2 in detail. An important role is played in this dimension by the standard fundamental domain. Definition 2.5.1. The standard fundamental domain, denoted D, is the set 1 D = z C z 1, (z) , (z) > 0 (2.5.2) ∈ | | ≥ |< | ≤ 2 =   cf. [Ser73, p. 78].

The domain D a fundamental domain for the action of P SL(2, Z) in the upperhalf plane of C. 2.5. HERMITE AND BERGE-MAR´ TINET CONSTANTS 17

Lemma 2.5.2. Let b = 2. Then we have the following value for the Hermite constant: γ = 2 = 1.1547.... The corresponding optimal 2 √3 lattice is homothetic to the Z-span of cube roots of unity in C. Proof. Consider a lattice L C = R2. Clearly, multiplying L by nonzero complex numbers does not⊂ change the value of the quotient λ (L)2 1 . area(C/L)

Choose a “shortest” vector z L, i.e. we have z = λ1(L). By re- 1 ∈ | | placing L by the lattice z− L, we may assume that the complex num- ber +1 C is a shortest element in the lattice. We will denote the new ∈ lattice by the same letter L, so that now λ1(L) = 1. Now complete +1 to a Z-basis τ, +1 { } for L. Thus τ λ1(L) = 1. Consider the real part (τ). We can adjust the basis| | b≥y adding or subtracting a suitable integer< to τ, so as to satisfy the condition 1 1 (τ) . −2 ≤ < ≤ 2 Thus, the basis vector τ may be assumed to lie in the standard funda- √3 mental domain (2.5.2). Then the imaginary part satisfies (τ) 2 , with equality possible precisely for = ≥

i π i 2π τ = e 3 or τ = e 3 . Moreover, if τ = r exp(iθ), then (τ) √3 sin θ = = . τ ≥ 2 | | The proof is concluded by calculating the area of the parallelogram in C spanned by τ and +1; λ (L)2 1 2 1 = . area(C/L) τ sin θ ≤ √3 | | 

Example 2.5.3. In dimensions b 3, the Hermite constants are harder to compute, but explicit values (as≥ well as the associated critical 1 lattices) are known for small dimensions, e.g. γ3 = 2 3 = 1.2599..., while γ4 = √2 = 1.4142.... 18 2. LINEAR ALGEBRA, LATTICES, CURVES

r

1 Figure 2.6.1. Circle of radius r has curvature k = r

A related constant γb0 , called the Berg´e-Martinet constant, is defined as follows: b γ0 = sup λ (L)λ (L∗) L (R , ) , (2.5.3) b 1 1 ⊆ k k Rb where the supremum is extended over all lattices L in . 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 (5.1.1). h i

2.6. Curves and their curvature We review the standard calculus topic of the curvature of a curve, which is indispensable to understanding principal curvatures of a sur- face (cf. Theorem 3.7.7 and Theorem 3.9.4). Remark 2.6.1. The second derivative of a function f(x) may be computed by differentiating f twice. Alternatively, f(x + h) + f(x h) 2f(x) f 00(x) = lim − − , h 0 2 → h which is true also for vector-valued functions (cf. Definition 2.6.6). Consider a parametrized curve α(t) = (α1(t), α2(t)) in the plane. Denote by C the underlying geometric curve : C = (x, y) ( t), x = α1(t), y = α2(t) . { | ∃ } dα Definition 2.6.2. We say α is a unit speed curve if dt = 1, 1 2 2 2 i.e. dα + dα = 1. dt dt

When
dealing
with a unit speed curve, it is customary to denote the parameter (called arclength) by s. 2.6. CURVES AND THEIR CURVATURE 19

s s Example 2.6.3. Let r > 0. Then the curve α(s) = r cos r , r sin r is a unit speed parametrisation of the circle of radius r, see Figure 2.6.1. Indeed, we have

dα 1 s 2 1 s 2 s s = r sin + r cos = sin2 + cos2 = 1. ds r − r r r r r s     r

Our main interest will be in space curves, therefore in the definition below we do not concern ourselves with the sign of the curvature of plane curves, either.

Definition 2.6.4. The (geodesic) curvature kα(s) 0 of a unit speed curve α(s) is defined by ≥ d2α k (s) = . α ds2

Example 2.6.5. For the circle of radius r parametrized as above, we have d2α 1 s 1 s = r cos , r sin ds2 r2 − r r2 − r      at s = 0, and so the curvature satisfies 1 s s 1 k = cos , sin = . α r − r − r r  

Note that in this case, the curvature is indep endent of the point, i.e. is a constant function of s. Definition 2.6.6. The osculating circle to the curve parametrized by α at the point α(s) is the limit of circles passing through the three points α(s), α(s h), α(s + h) as h 0. − → Note that the osculating circle and the curve are ”better than tan- gent” (they have second order tangency). Since the second derivative is computed from the same triple of points for α(s) and for the osculating circle (cf. Remark 2.6.1), we have the following [We55, p. 13]. Theorem 2.6.7. The curvatures of the osculating circle and the curve at the point of tangency are equal. Example 2.6.8. 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 the intersection of the perpendicular bisector of OB with the y-axis. Let D = (0, x), cf. Figure 2.6.2. 20 2. LINEAR ALGEBRA, LATTICES, CURVES

C

A B

O D

Figure 2.6.2. Finding the osculating circle of a parabola

1 2 2 2 OA 2 √x +(ax ) Triangle OAC yields sin ψ = OC = r , triangle OBD 2 1 √x2+a2x4 2 yields sin ψ = BD = ax , and 2 = ax , so that 1 (x2 + OB √x2+a2x4 r √x2+a2x4 2 2 4 2 1 2 2 1+a2x2 a x ) = arx , and 2 (1 + a x ) = ar, so that r = 2a . 1 1 Taking the limit as x 0, we obtain r = , hence k = = 2a = → 2a r f 00(0). Thus the curvature of the parabola at its vertex equals the second derivative with respect to x (even though x is not the arclength parameter of the graph).

Theorem 2.6.9. Let x0 be a critical point of f(x). Then the cur- vature of the graph of f at (x0, f(x0)) equals

k = f 00(x ) . | 0 | Proof. We would like to parametrize the graph by α(s) = (s, f(s)). Then we have α00(s) = (0, f 00(s)) and α00(s) = f 00(s) . However, (s, f(s)) is not a unit speed parametrisation of| the graph.| | Nonetheless,| the sec- ond order Taylor expansion of f at x0 clearly has the same osculating circle as f, and we conclude the argument as for ax2. 

Theorem 2.6.10. Suppose a curve α(t) satisfies α0(t) = 0 at every point. Then there exists a unit speed parametrition, define6 d by equa- tion (2.6.1) below, of the underlying geometric curve C.

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

b 2 L = 1 + (f 0(x)) dx . a Z p 2.7. GREAT CIRCLES AND CLAIRAUT’S RELATION 21

The element of length ds decomposes by Pythagoras’ theorem as fol- lows: ds2 = dx2 + dy2. More generally, for α(t) = (α1(t), α2(t)), we have

2 2 b d(α1) dα2 b dα L = + dt = dt. v a dt ! dt ! a dt Z u Z u We set t t dα s(t) = dτ. (2.6.1) a dτ Z ds dα By the Fundamental Theorem of Calculus, = . Then by chain dt dt rule dα dα dt dα 1 dα 1 = = = . ds dt ds dt ds/dt dt dα/dt dα | | Thus ds = 1. This is called the arc length parametrisation of the geometric curve C. 

1 2 3/2 Example 2.6.11. Let f(x) = 3 (2+x ) . Find arc length parametri- sation of the graph of f. Remark 2.6.12 (Curves in R3). A space curve may be written in coordinates as α(s) = (α1(s), α2(s), α3(s)). 2 dα 3 dαi Here s is the arc length if ds = 1 i.e. i=1 ds = 1.   Example: helix α(t) = (a cos ωt, a sinPωt, bt).

(i) make a drawing in case a = b = ω = 1. (ii) parametrize by arc length. (iii) compute the curvature.

2.7. Great circles and Clairaut’s relation To provide a geometric feeling for the geodesic equation which we will encounter in section 3.5, we will establish a connection between this system of nonlinear second order differential equations, on the one hand, and spherical great circles, on the other. Such a connection is es- tablished via the intermediary of Clairaut’s relation (cf. Theorem 2.7.10). In this section, we will verify Clairaut’s relation synthetically for great circles, and also show that the latter satisfy a first order differential equation. In section 3.5 we will complete the connection by deriving Clairaut’s relation from the geodesic equation. 22 2. LINEAR ALGEBRA, LATTICES, CURVES

Let S2 R3 be the unit sphere defined by the equation x2+y2+z2 = 1. A plane⊂P through the origin is given by an equation ax+by+cz = 0, a 0 where the vector b = 0 is nonzero. A great circle G of S2 is c 6 0 given by an intersection  G= S2 P . ∩ Example 2.7.1 (A parametrisation of the equator of S2).

α(t) = (cos t)e1 + (sin t)e2. In general, every great circle can be parametrized by α(t) = (cos t)v + (sin t)w where v and w are orthonormal.

Lemma 2.7.2. Scalar product of vector valued functions satisfies Leibniz’s rule:

f, g 0 = f 0, g + f, g0 . (2.7.1) h i h i h i Proof. See [Leib].  Lemma 2.7.3. Let α : R S2 be a parametrized curve on S2 R3. dα → ⊂ Then the tangent vector is perpendicular to the position vector α(t): dt dα α(t), = 0. dt D E d Proof. We have α(t), α(t) = 1 by definition of S2. We apply h i dt d to obtain α(t), α(t) = 0. Next, we apply Leibniz’s rule (2.7.1) to dth i obtain dα dα dα d α(t), + , α(t) = 2 α(t), = (1) = 0. dt dt dt dt D E D E D E  2.7. GREAT CIRCLES AND CLAIRAUT’S RELATION 23

Definition 2.7.4. A spherical triangle is the following collection of 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 vectors to the sides. Here we assume that sides haves length < π, while the three vertices do not lie on a great circle. Remark 2.7.5 (Spherical sine law). sin a sin b sin c = = sin α sin β sin γ

π Example 2.7.6. Let γ = , then sin a = sin c sin γ. 2 Note that for small values of a, c we recapture the Euclidean formula a = c sin α as the limiting case:

Definition 2.7.7. A longitude (meridian, kav orech) is a great circle passing through the North Pole e3 = (0, 0, 1). 24 2. LINEAR ALGEBRA, LATTICES, CURVES

Lemma 2.7.8. Assume G is not a longitude (meridian) and let p G be the point with the maximal z-coordinate among the points of G∈. Then G is perpendicular at p to the longitude (meridian) passing through p. Proof. Let α(t) be a parametrisation of G with α(0) = p. Since the function α(t), e achieves its maximum at t = 0, we have h 3i d α(t), e3 = 0. dt 0 h i

By Leibniz’s rule,

dα de3 , e3 = α(t), = 0, dt 0 − dt D E D E . dα . . since e is constant. Let α(t) = . Thus α(0), e = 0. Also α(0), p = 3 dt h 3i h i 0 by Lemma 2.7.3. Since the tangent vector to the longitude (meridian) through p lies in the plane spanned by p and e3, the lemma follows.  Definition 2.7.9. Spherical coordinates ρ, θ, ϕ. Here ρ = x2 + y2 + z2 = √r2 + z2 is the distance to the origin,p while ϕ is the angle with the z-axis, so that cos ϕ = z, and θ is the angle inherited from polar coordinates in y the x, y plane, so that tan θ = x . A latitude (kav rochav) on S2 is a circle satisfying the equation ϕ = constant. It can be parametrized by setting θ(t) = t, ϕ(t) = constant. Note that a latitude is not a great circle unless it is the equator ϕ = 0. Theorem 2.7.10 (Clairaut’s relation). Let α(t) be a parametrized great circle on S2 R3. Let r(t) denote the distance from α(t) to . the z axis, and let⊂γ(t) denote the angle between α(t) and the vector tangent− to the latitude at the point α(t). Then r(t) cos γ(t) = const.

Here the constant has value const = r min, where r min is the least dis- tance from a point of G to the z-axis.

π . Proof. Note that we have an angle of 2 γ between α(t) and the vector tangent to the longitude (meridian) at−α(t) (this is equivalent to the vanishing of the metric coefficient g12 for a surface of revolution). Let p S2 be the point of α(t) with maximal z-coordinate. Note that if ϕ(t∈) is the spherical coordinate ϕ of α(t), then r(t) = sin ϕ(t). Consider the spherical triangle with vertices α(t), p, and e3 : 2.7. GREAT CIRCLES AND CLAIRAUT’S RELATION 25 π By Lemma 2.7.8, the angle at p is . Hence by the law of sines, 2 π sin c sin( γ) = sin b 2 − where c = ϕ(t) = arc of longitude (meridian) joining e3 and α(t), and b = arc of longitude (meridian) joining p and e3. Since b is inde- pendent of t, the theorem is proved.  Assume G is not a longitude (meridian), so we can parametrize it by the value of the spherical coordinate θ. Since ρ 1, we think of G as defined by a function ϕ = ϕ(θ). ≡ Theorem 2.7.11. The great circle G satisfies the differential equa- tion 1 1 dϕ 2 1 + = , (2.7.2) r2 r4 dθ const2   where r = sin ϕ and const = sin ϕ min from Theorem 2.7.10. Proof. An ‘element of length’, ds, along C decomposes into a lon- gitudinal (along a longitude (meridian), north-south) displacement dϕ, and a latitudinal (east-west) displacement rdθ. These are related by ds sin γ = dϕ ds cos γ = rdθ dϕ so that ds2 = (dϕ)2 + (rdθ)2. Hence rdθ tan γ = dϕ, or tan γ = . rdθ 1 Hence cos2 γ = . Therefore by Theorem 2.7.10, we obtain dϕ 2 1 + rdθ const
2 dϕ 2 1 + = 1, r rdθ     ! or 1 1 dϕ 2 1 + = where r = sin ϕ. r2 r4 dθ const2   This equation is solved explicitly in terms of integrals in Example 3.5.2 below. 

CHAPTER 3

Local geometry of surfaces

The differential geometry of surfaces in Euclidean 3-space starts with the observation that they inherit a metric structure from the am- bient space. We would like to understand which geometric properties of this structure are intrinsic, in a sense to be clarified. Remark 3.0.12. Following [Ar74, Appendix 1, p. 301], note that a piece of paper may be placed flat on a table, or it may be rolled into a cylinder, or it may be rolled into a cone. However, it cannot be transformed into the surface of a sphere, that is, without tearing or stretching. Understanding this phenomenon quantitatively is our goal, cf. Figure 4.3.1.

3.1. First fundamental form 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 it allows one to measure the length of curves on the surface. Consider a parametrized surface in R3 given by a map x(u1, u2) or x : R2 R3. (3.1.1) →

We will always assume that x is differentiable. Denote by Jx its Jaco- bian matrix, i.e. the 3 2 matrix × dxi J = , (3.1.2) x duj   where xi are the components of the vector valued function x. Definition 3.1.1. A surface x is called regular if one of the follow- ing equivalent conditions is satisfied: ∂x ∂x (1) the vectors ∂u1 and ∂u2 are linearly independent; (2) the Jacobian matrix (3.1.2) is of rank 2.

27 28 3. LOCAL GEOMETRY OF SURFACES

Let , denote the inner product in R3. For i = 1, 2 and j = 1, 2, h i 1 2 define functions gij(u , u ) by ∂x ∂x g (u1, u2) = , . (3.1.3) ij ∂ui ∂uj D E ∂x Definition 3.1.2. The vectors x = are called the tangent i ∂ui vectors to the surface.

Thus the matrix (gij) is the Gram matrix of the pair of tangent vectors: T (gij) = Jx Jx, cf. formula (2.4.2). Definition 3.1.3. The first fundamental form of the surface x is the bilinear form on the tangent plane defined by the restriction of the ambient inner product. With respect to the basis x , x , it is given { 1 2} by the matrix (gij), where g = x , x . ij h i ji The coefficients gij are sometimes called ‘metric coefficients.’

Remark 3.1.4. We have gij = gji as the inner product is symmet- ric. Example 3.1.5. Let x(u1, u2) = (u1, u2, 0) R3. This is a parametri- 3 ∈ t t sation of the xy-plane in R . Then x1 = (1, 0, 0) and x2 = (0, 1, 0) . Then 1 1 g11 = x1, x1 = 0 , 0 = 1, h i *0 0+ 1 0    etcetera. Thus we have (g ) = = I. ij 0 1   Example 3.1.6. Let x(u1, u2) = (cos u1, sin u1, u2). This formula 1 1 t provides a parametrisation of the cylinder. We have x1 = ( sin u , cos u , 0) t − and x2 = (0, 0, 1) , while sin u1 sin u1 − 1 − 1 2 2 g11 = cos u , cos u = sin + cos = 1, * 0   0 +     1 0 etcetera. Thus (g ) = = I. ij 0 1   3.1. FIRST FUNDAMENTAL FORM 29

Remark 3.1.7. The first fundamental form does not contain all the information (even up to Euclidean congruences) about the surface. Indeed, the plane and the cylinder have the same first fundamental form. Example 3.1.8 (Surfaces of revolution). Here it is customary to use the notation u1 = θ and u2 = ϕ. The starting point is a curve C in the xz-plane, parametrized by a pair of functions x = f(ϕ), z = g(ϕ). The surface of revolution (around the z-axis) defined by C is parametrized as follows : x(θ, ϕ) = (f(ϕ) cos θ, f(ϕ) sin θ, g(ϕ)). If we start with the vertical line f(ϕ) = 1, g(ϕ) = ϕ, the resulting surface of revolution is the cylinder. For f(ϕ) = sin ϕ and g(ϕ) = cos ϕ, we obtain the sphere S2 in spherical coordinates. Let us calculate the first fundamental form of a surface of revolution:

∂x x = = ( f sin θ, f cos θ, 0), 1 ∂θ − ∂x df df dg x = = cos θ, sin θ, 2 ∂ϕ dϕ dϕ dϕ   f sin θ 2 − 2 2 2 2 2 g11 = f cos θ = f sin θ + f cos θ = f

0

2 df cos θ dϕ df 2 dg 2 df 2 dg 2 df sin θ 2 2 g22 =  dϕ  = (cos θ + sin θ) + = + dg dϕ dϕ dϕ dϕ          dϕ    df f sin θ cos θ dϕ df df −f cos θ df sin θ g12 = ,  dϕ  = f sin θ cos θ + f cos θ sin θ = 0. * 0  dg + − dϕ dϕ dϕ     Thus   f 2 0 2 2 (gij) = df dg . 0 dϕ + dϕ ! Thus for S2,     sin2 ϕ 0 (g ) = . ij 0 1   30 3. LOCAL GEOMETRY OF SURFACES

Example 3.1.9. The pseudosphere (so called because its Gaussian curvature equals 1) is the surface of revolution defined by the func- − ϕ tions f(ϕ) = eϕ and g(ϕ) = 1 e2ψdψ, where < ϕ 0. 0 − −∞ ≤ 2ϕ Z Then g11 = e , while p

g = (eϕ)2 + (√1 e2ϕ)2 22 − = e2ϕ + 1 e2ϕ = 1. − e2ϕ 0 Thus (g ) = . ij 0 1   Exercise 3.1.10. Compute the coefficients gij for the standard parametrisation of the graph of z = f(x, y) by (u1, u2, f(u1, u2)).

3.2. Length and area Let us now explain how to measure the length of curves on a surface in terms of the metric coefficients of the surface. Let α : [a, b] R2, α(t) = (α1(t), α2(t)) → be a plane curve. Let β(t) = x α(t) be a curve on the surface x. The length of β is calculated as follo◦ws using chain rule: b dβ b 2 ∂x dαi L = dt = dt dt ∂ui dt a a i=1 Z Z X 1/2 b dαi dαj = x , x dt i dt j dt Za   b dαi dαj 1/2 = x , x dt = h i ji dt dt Za   b dαi dαj = g (α(t)) dt. ij dt dt Za r Thus b dαi dαj L = g (α(t)) dt. (3.2.1) ij dt dt Za r Note that if gij = δij is the identity matrix, then the length of the curve β = x α on the surface equals the length of the curve α of R2. ◦ b dα1 2 dα2 2 Indeed, If gij = δij, then L = + dt = the length a s dt dt Z     of α(t) in R2. 3.3. CHRISTOFFEL SYMBOLS 31

Definition 3.2.1 (Computation of area). The area of the surface x is computed by integrating the expression

1 2 det(gij)du du (3.2.2)

q 1 2 over the domain U of the map x. Thus area(x) = U det(gij)du du . The presence of the square root in the formulaR ispexplained in in- finitesimal calculus in terms of the Gram matrix, cf.

3.3. Christoffel symbols The Christoffel symbols, roughly speaking, account for how the sur- face twists in space. They are, however, coordinate dependent and do not have an intrinsic geometric meaning. We will see that the Christof- fel symbols also control the behavior of geodesics on the surface, i.e. curves that are to the surface what straight lines are to a plane, or what great circles are to a sphere, cf. Definition 3.4.1. ∂x Let x : R2 R3 be a parametrized surface, and let x = . The → i ∂ui normal vector n(u1, u2) to the surface at the point x(u1, u2) R3 is defined in terms of the vector product, cf. Definition 2.1.11, as ∈follows: x x n = 1 × 2 , x x | 1 × 2| so that n, x = 0 i. h ii ∀

3 The vectors x1, x2, n form a basis for R (since x is regular by hypothe- sis). Recall that the gij were defined in terms of first partial derivatives k of x. Meanwhile, the Christoffel symbols Γij are defined in terms of the second partial derivatives (since they are not even tensors, one does not stagger the indices). Namely, they are the coefficients of the decomposition of the second partial derivative vector xij, defined by ∂2x x = ij ∂ui∂uj with respect to the basis (x1, x2, n) : 32 3. LOCAL GEOMETRY OF SURFACES

k Definition 3.3.1. The Christoffel symbols Γij are uniquely deter- mined by the formula 1 2 xij = Γijx1 + Γijx2 + Lijn

(the coefficients Lij will be discussed in section 10). Proposition 3.3.2. We have the following formula for the Christof- fel symbols: Γk = x , x g`k, where (gij) is the inverse matrix of g . ij h ij `i ij Proof. We have x , x = Γk x + L n, x = Γk x , x + h ij `i h ij k ij `i h ij k `i L n, x = Γk g . We now multiply by g`m and sum: h ij `i ij k` x , x g`m = Γk g g`m = Γk δm = Γm. h ij `i ij k` ij k ij This is equivalent to the desired formula. 

k k k Remark 3.3.3. We have the following relation: Γij = Γji, or Γ[ij] = 0 ijk. ∀ Example 3.3.4. Calculation of the Christoffel symbols for the plane x(u1, u2) = 1 2 (u , u , 0). We have x1 = (1, 0, 0), x2 = (0, 1, 0). Thus xij = 0 i, j. Hence Γk 0 i, j, k. ∀ ij ≡ ∀ Example 3.3.5. Calculation of the symbols for the cylinder x(u1, u2) = 1 1 2 1 1 (cos u , sin u , u ). The normal vector is n = (cos u , sin u , 0), while x1 = 1 1 k ( sin u , cos u , 0), x2 = (0, 0, 1). Thus x22 = 0, x21 = 0 and so Γ22 = 0 − k k and Γ12 = Γ21 = 0 k. ∀ 1 2 Meanwhile, x11 = ( cos u , sin u , 0). This is proportional to n : x = 0x + 0x + ( 1)−n. Hence− Γk = 0 k. 11 1 2 − 11 ∀ ∂ Definition 3.3.6. We write g = (g ). ij;k ∂uk ij Lemma 3.3.7. In terms of the symmetrisation notation introduced in Remark 2.1.5, we have m gij;k = 2gm iΓj k. { } k m Proof. Indeed, gij;k = ∂u xi, xj = xik, xj + xi, xjk = Γikxm, xj + m h i h i h i h i Γjkxm, xi . By definition of the metric coefficients, we have gij;k = h m im m m m m  Γikgmj + Γjkgmi = gmjΓik + gmiΓjk = 2gm jΓi k or 2gm iΓj k. { } { } Theorem 3.3.8. The Christoffel symbols can be expressed in terms of the first fundamental form and its derivatives as follows : 1 Γk = (g g + g )g`k, ij 2 i`;j − ij;` j`;i ij where g is the inverse matrix of gij. 3.3. CHRISTOFFEL SYMBOLS 33

Γ1 ij j = 1 j = 2

i = 1 0 1 df f dϕ 0 i = 2 1 df f dϕ k Table 3.3.1. Symbols Γij of a surface of revolution (3.3.1)

Proof. Applying Lemma 3.3.7 three times, we obtain m m m gi`;j gij;` + gj`;i = 2gm iΓ` j 2gm iΓj ` + 2gm jΓ` i { } { } { } − m − m m m m m = gmiΓ`j + gm`Γij gmiΓj` gmjΓi` + gmjΓ`i + gm`Γji m − − = 2gm`Γji . 1 Thus (g g +g )g`k = Γmg g`k = Γmδk = Γk , as required.  2 i`;j− ij;` j`;i ij m` ij m ij 1 1 Lemma 3.3.9. For a surface of revolution we have Γ11 = Γ22 = 0, f df 1 dϕ while Γ12 = f 2 , cf. Figure 3.3.1.

Proof. For the surface of revolution x(θ, ϕ) = (f(ϕ) cos θ, f(ϕ) sin θ, g(ϕ)), (3.3.1) the metric coefficients are given by the matrix f 2 0 2 2 (gij) = df dg . 0 dϕ + dϕ !

Since the off-diagonal coefficientsg12 = 0 vanish, the coefficients of the inverse matrix satisfy 1 gii = . (3.3.2) gii ∂ We have (g ) = 0 since g depend only on ϕ. Thus the terms ∂θ ii ii gii;1 = 0 (3.3.3) 1 vanish. Let us now compute the Christoffel symbols Γij for k = 1. Using formulas (3.3.2) and (3.3.3), we obtain

1 1 Γ11 = (g11;1 g11;1 + g11;1) by formula (3.3.2) 2g11 − = 0 by formula (3.3.3). 34 3. LOCAL GEOMETRY OF SURFACES

Γ1 Γ2 ij j = 1 j = 2 ij j = 1 j = 2

λ2 λ1 i = 1 λ1 2λ i = 1 λ2 2λ 2λ −2λ λ2 λ1 i = 2 2λ λ1 i = 2 2λ λ2 − 2λ 2λ k Table 3.3.2. Symbols Γij of a conformal metric λδij

Similarly, 1 1 g11;2 Γ12 = (g11;2 g12;1 + g12;1) = 2g11 − 2g11 d (f 2) f df = dϕ = dϕ , 2f 2 f 2 d (0) 1 1 g12;2 dϕ while Γ = (g12;2 g22;1 + g12;2) = = = 0.  22 2g11 − g11 g11 1 2 1 λ1 1 λ1 Lemma 3.3.10. For gij = λ(u , u )δij we have Γ11 = 2λ , Γ22 = −2λ , 1 λ2 and Γ12 = 2λ (see Table 3.3.2).

Proof. We have 1 Γ1 = (g g + g ), ij 2λ i1;j − ij;1 j1;i and the lemma follows by examining the cases. 

3.4. What is a geodesic on a surface? It is the path of an ant crawling along the surface of an apple, ac- cording to Gravitation, a Physics textbook [MiTW73, p. 3]. Imagine that we peel off a narrow strip of the apple’s skin along the ant’s tra- jectory, and then lay it out flat on a table. What we obtain is a straight line, revealing the ant’s ability to travel along the shortest path. On the other hand, a geodesic is defined by a certain nonlinear sec- ond order ordinary differential equation, cf. (3.4.1). To make the geo- desic equation more concrete, we will examine the case of the surfaces of revolution. Here the geodesic equation transforms into a conserva- tion law called Clairaut’s relation. The latter lends itself to a synthetic verification for spherical great circles, as in Theorem 2.7.10. 3.4. WHAT IS A GEODESIC ON A SURFACE? 35

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’s foundational work in this area. Namely, Morse systematically used the length functional on the space of curves. The simple idea of using the energy functional instead of the length functional was not exploited until later. The use of energy simplifies calculations considerably, as we will see in Section 3.8. Consider a plane curve R R2 where α = (α1(s), α2(s)). If x : s →α (u1,u2) R2 R3 is a surface, the composition → R R2 R3 s →α (u1,u2) →x yields a curve β = x α. ◦ Definition 3.4.1. A curve β = x α is a geodesic on the surface x if the one of the following two equivalen◦ t conditions is satisfied: (a) we have for each k = 1, 2,

00 0 0 0 d (αk) + Γk (αi) (αj) = 0 where = , (3.4.1) ij ds meaning that d2αk dαi dαj ( k) + Γk = 0; ∀ ds2 ij ds ds (b) the vector β00 is perpendicular to the surface and one has i j β00 = Lijα 0α 0n. (3.4.2) Remark 3.4.2. The equations (3.4.1) will be derived using the calculus of variations, in Section 3.8. Furthermore, by Lemma 3.4.4, such a curve β must have constant speed.

i Proof of equivalence. Write β = x α, then β0 = xiα 0. Fur- thermore, ◦

d i i j i k β00 = (x α)α 0 + x α 00 = x α 0α 0 + x α 00. ds i ◦ i ij k Now use geodesic equation αk00 = Γk αi0αj0. Thus − ij i j k i j β00 = x α 0α 0 Γ α 0α 0x . ij − ij k k Meanwhile xij = Γijxk + Lijn thus k i j i j k i j i j β00 = Γ α 0α 0x + L α 0α 0n Γ α 0α 0x = L α 0α 0n, ij k ij − ij k ij proving (3.4.2).  36 3. LOCAL GEOMETRY OF SURFACES

k Example 3.4.3. In the plane, all coefficients Γij = 0 vanish. Then 00 the equation becomes (αk) = 0, i.e. α(s) is a linear function of s. In other words, the graph is a straight line. Lemma 3.4.4. A curve β satisfying the geodesic equation (3.4.1) is necessarily constant speed.

d 2 Proof. From formula (3.4.2), we have ( β0 ) = 2 β00, β0 = ds k k h i L αi0αj0αk0 n, x = 0.  ij h ki 3.5. Geodesics on a surface of revolution Theorem 3.5.1. Let u1 = θ, u2 = ϕ. Let x = f(ϕ) cos θ, y = f(ϕ) sin θ, and z = g(ϕ) be the components of a surface of revolution. Then the geodesic equation for k = 1 is equivalent to Clairaut’s rela- tion r cos γ = const (cf. Theorem 2.7.10). Proof. The Christoffel symbols for k = 1 are given by Lemma 3.3.9. We will use the shorthand notation θ(s), ϕ(s) respectively for α1(s), α2(s). d Let 0 = . The equation of geodesic β = x α for k = 1 becomes ds ◦ 1 0 = θ00 + 2Γ12θ0ϕ0 df 2f dϕ 0 0 = θ00 + θ ϕ f 2

2 df 0 0 = f θ00 + 2f θ ϕ dϕ 0 0 = (f 2θ ) , and therefore 0 f 2θ = const. Since the curve β is constant speed by Lemma 3.4.4, we can assume the paramenter s is arclength. Now the angle γ between β and the latitude satisfies x dβ ∂x cos γ = 1 , where x = (θ, ϕ). x ds 1 ∂θ | 1|  Thus 0 0 1 0 0 θ ϕ cos γ = x1, x1θ + x2ϕ = x1, x1 + x1, x2 x1 h i x1 h i x2 h i 2 | | | | x1 | | g12=0 chain rule | | 0 0 0 = θ x1 = |fθ ={zrθ .} | | | {z } | {z } 0 Thus r cos γ = r2θ = const, proving Clairaut’s relation.  3.6. WEINGARTEN MAP 37

In the case of a surface of revolution, the geodesic equation can be integrated by means of primitives :

dx dx 0 0 0 0 1 = , = x θ + x ϕ , x θ + x ϕ ds ds h 1 2 1 2 i  

2 2 0 0 0 df dg 0 = g (θ )2 + g (ϕ )2 = f 2(θ )2 +  +  (ϕ )2. 11 22 dϕ dϕ       g22      dθ 2 dϕ 2 | {z } ds 2 Thus, 1 = f 2 + g . Now multiply by : ds 22 ds dθ       ds 2 dϕ ds 2 dϕ 2 = f 2 + g = f 2 + g . dθ 22 ds dθ 22 dθ       ds f 2 Now from Clairaut’s relation we have dθ = c , hence we obtain the f 4 2 dϕ 2 formula c2 = f + g22 dθ or 1
1 g dϕ 2 = + 22 c2 f 2 f 4 dθ   which is the spherical equation (2.7.2)! Furthermore, we have

4 f 2 2 2 dϕ c2 f f f c = − = 2 − 2 . dθ g c df dg s 22 v + u dϕ dϕ u t    Thus 2 2 df dg 1 dϕ + dϕ θ = c v dϕ + const. f u f 2 c2  Z u t − Example 3.5.2. On S2, we have f(ϕ) = sin ϕ, g(ϕ) = cos ϕ, thus we obtain dϕ θ = c + const, sin ϕ sin2 ϕ c2 Z − which is the equation of a greatpcircle in spherical coordinates.

3.6. Weingarten map Definition 3.6.1. Given a function f of several variables, its di- rectional derivative f, in the direction of a vector v arising as the ∇v 38 3. LOCAL GEOMETRY OF SURFACES

dc velocity vector v = dt of a curve c(t), is defined by setting d (f c(t)) f = ◦ . ∇v dt Lemma 3.6.2. The definition of directional derivative is indepen- dent of the choice of the curve c representing the vector v. Proof. The lemma is proved in elementary calculus.  Let p = x(u1, u2) be a point of a surface in R3. Definition 3.6.3. The tangent plane to the surface x at the point p is denoted Tp and is defined “naively” to be the plane passing through p and spanned by vectors x1 and x2, or alternatively as the plane per- pendicular to the normal vector n at p. Thus we have an orthogonal decomoposition

R3 = T ⊥ Rn. p ⊕ Example 3.6.4. Suppose x is a parametrisation of S2 R3 (unit sphere). At a point (a, b, c) S2, the normal vector is the⊂ position a ∈ vector itself: n = b . c Now let us return to the set-up R R2 R3. t →α (u1,u2) →x dβ Let β = x α. Let v Tp be defined by v = dt t=0. By chain rule, i ◦ ∈ v = dα x . The normal vector n α(t) along β(t) varies from point to dt i point. Thecurve β represents the◦class of curves with initial vector v.

Proposition 3.6.5. Let p M, and v TpM. The directional derivative n satisfies ∈ ∈ ∇v d (n α(t)) n = ◦ . (3.6.1) ∇v dt Proof. We extend n to a vector field N(x, y, z) defined in an open neighborhood of p R3, so that we have n(u1, u2) = N (x(u1, u2)), and in particular n(∈α(t)) = N(β(t)), where β = x α. The left hand ◦ side of (3.6.1) should be understood in the sense of vN. Then by Lemma 3.6.2, ∇ d (N β(t)) d (n α(t)) N = ◦ = ◦ , ∇v dt dt proving the proposition.  3.6. WEINGARTEN MAP 39

Definition 3.6.6. The Weingarten map (shape operator) W : T T p → p is the endomorphism of the tangent plane given by directional deriva- tive of n: d W (v) = n = n α(t), ∇v dt ◦ t = 0 where β = x α is chosen so that β(0) = p while β0(0) = v. ◦ By Leibniz’s rule, 1 d W (v), n = n, n = n α(t), n α(t) = 0, h i h∇v i 2 dth ◦ ◦ i and therefore W (v) indeed lies in Tp, so that the map W is well defined. 1 2 1 2 Example 3.6.7. Plane x(u , u ) = (u , u , 0). x1 = e1, x2 = e2, n = x1 x2 × = e1 e2 = e3 constant ! Thus vn 0 and W (v) 0, and (x1 x2) × ∇ ≡ ≡ the ×Weingarten map is identically zero. Lemma 3.6.8. The Weingarten map of the sphere of radius r > 0 1 is given by r Id. 1 Proof. We have n α(t) = β(t). Hence ◦ r W (v) = n α(t) ∇v ◦ d = n α(t) dt ◦ t = 0

1 d = β(t) r dt t = 0 1 = v. r 1 1 Thus W (v) = v, and W = Id.  r r Example 3.6.9. For the cylinder, we have the parametrisation x(u1, u2) = (cos u1, sin u1, u2), and n = (cos u1, sin u1, 0). Thus, ∂ n = n = ( sin u1, cos u1, 0) ∇x1 ∂u1 − ∂ n = n = (0, 0, 0). ∇x2 ∂u2 40 3. LOCAL GEOMETRY OF SURFACES

i Let v = v xi. Then sin u1 1 1 − 1 i vn = v xi n = v x1 n = v cos u , ∇ ∇ ∇  0  and therefore,   sin u1 1 − 1 1 W (v) = v cos u = v x1,  0  i   where v = v xi. Thus, W (x1) = x1, while W (x2) = 0.

Since (x1, x2) is a basis of Tp, we may make the following definition. i Definition 3.6.10. The coefficients L j of the Weingarten map are defined by i W (xj) = L jxi. Example 3.6.11. For the plane, we have (Li ) 0, for the sphere, j ≡ 1 0 1 (Li ) = r = δi . j 0 1 r j  r  1 For the cylinder, we have L 1 = 1. The remaining coefficients vanish, so that 1 0 (Li ) = . j 0 0   Theorem 3.6.12. The operator W is selfadjoint.

Proof. As in the definition of the Weingarten map, vn is the derivative of n along a curve with initial vector v. Thus if γ(∇t) = x(t, a) then γ0(t) = x1. ∂n Therefore 1 = n by definition of the directional derivative. ∂u ∇x1 Thus the coordinate lines x(u1, a) and x(b, u2) of the chart (u1, u2) ∂ allow us to write ∂uj n = xj n. Therefore we have ∇ ∂n W (x ), x = , x h i ji ∂ui j   ∂ ∂ = n, x n, x ∂ui h ji − h ∂ui ji ∂2x = n, , − ∂ui∂uj   which is an expression symmetric in i and j.  3.7. GAUSSIAN CURVATURE 41

3.7. Gaussian curvature We continue with the terminology and notation of the previous section. Definition 3.7.1. The Gaussian curvature function K = K(u1, u2) of the surface x is the determinant of the Weingarten map : K = det(Li ) = L1 L2 L1 L2 = 2L1 L2 j 1 2 − 2 1 [1 2] (the skew-symmetrisation notation was defined in 2.1.5). Example 3.7.2. Cylinder, plane K = 0, cf. Figure 4.3.1. For a 1 0 1 sphere of radius r, we have K = det r = . 0 1 r2  r  Definition 3.7.3. The second fundamental form IIp is the bilinear form on the tangent plane T defined for u, v T by p ∈ p II(u, v) = n, v . −h∇u i It may be helpful to keep in mind the observation that the second fundamental form measures the curvature of geodesics on x viewed as curves of R3 (cf. 2.6.4) as illustrated by Theorem 3.7.7 below.

Definition 3.7.4. The coefficients Lij of the second fundamental form are defined to be ∂n L = II(x , x ) = , x . ij i j − ∂ui j   Lemma 3.7.5. The coefficients Lij of the second fundamental form are symmetric in i and j, more precisely L = + x , n . ij h ij i Proof. We have n, x = 0. Hence ∂ n, x = 0, i.e. h ii ∂uj h ii ∂ n, x + n, x = 0 ∂uj i h iji   or n, x = n, x = +II(x , x ) h iji −h∇xj ii j i and the proof is concluded by the equality of mixed partials.  Lemma 3.7.6. The second fundamental form allows us to identify the normal component of the second partials of the map x, namely: xij = k Γijxk + Lijn.

k Proof. Let xij = Γijxk + cn and form the inner product with n to obtain L = x , n = 0 + c n, n = c.  ij h ij i h i 42 3. LOCAL GEOMETRY OF SURFACES

Theorem 3.7.7. Let β(s) be a unit speed geodesic on the surface in R3. Then II(β0, β0) = kβ(s), | | 3 where kβ is the curvature of β as a curve in R . Proof. We apply formula (3.4.2). Since n = 1, we have | | def i j k = β00 = L α 0α 0 . β | | | ij | Also i j i j i j II(β0, β0) = II(xiα 0, xjα 0) = α 0α 0II(xi, xj) = α 0α 0Lij, completing the proof.  Proposition 3.7.8. We have the following relation between the Weingarten map and the second fundamental form: L = Lk g . ij − j ki Proof. By definition, ∂ L = x , n = n, x = Lk x , x = Lk g . ij h ij i − ∂uj i −h j k ii − j ki    Theorem 3.7.9. We have the following three equivalent formulas for the Gaussian curvature: i 1 2 (a) K = det(L j) = 2L [1L 2]; (b) K = det(Lij ) ; det(gij ) (c) K = 2 L L2 . − g11 1[1 2] Proof. The first formula is our definition of K. To prove the second formula, we use Proposition 3.7.8. By the multiplicativity of determinant with respect to matrix multiplication,

i 2 det(Lij) det(L j) = ( 1) . − det(gij) Let us now prove formula (c). The proof is a calculation. Note that by definition, 2L L2 = L L2 L L2 . Hence 1[1 2] 11 2 − 12 1 2 2 1 1 2 2 1 2 2 L1[1L 2] = L 1g11 L 1g21 L 2 L 2g11 L 2g21 L 1 −g11 g11 − − −
1 1 2
2 2 1 2
2 2
= g11L 1L 2 g21L 1L 2 g11L 2L 1 + g21L 2L 1 g11 − − i
= det(L j) = K. k Note that we can either calculate using the formula Lij = L jgki, or k − the formula Lij = L igkj. Only the former one leads to the appropri- ate cancellations as−above.  3.8. CALCULUS OF VARIATIONS AND THE GEODESIC EQUATION 43

3.8. Calculus of variations and the geodesic equation Let α(s) = (α1(s), α2(s)), and consider the curve β = x α on the surface : ◦ x [a, b] α R2 R3. s → → Consider the energy functional

b 2 (β) = β0 ds. E k k Za d dβ 0 Here = . We have = x (αi) by chain rule. Thus 0 ds ds i b b b i j i j (β) = β0, β0 ds = xiα 0, xjα 0 ds = gijα 0α 0ds. (3.8.1) E a h i a a Z Z D E Z 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 , then for any perturbation δ vanishing at the endp◦ oints, the followingEderivative vanishes:

d 0 = (x (α + tδ)) dt E ◦ t = 0 b 0 0 d i i j j = gij(α + tδ ) (α + tδ ) ds (from equation (3.8.1)) dt a t = 0 Z  b 0 0 b 0 0 0 0 ∂ (gij (α + tδ)) i j i j j i = ◦ α α ds + gij α δ + α δ ds, ∂t Za Za A B
so that| we have {z } | {z } A + B = 0. (3.8.2) We will need to compute both the t-derivative and the s-derivative of the first fundamental form. The formula is given in the lemma below.

Lemma 3.8.1. The partial derivatives of gij = gij (α(s) + tδ(s)) along β = x α are given by the following formulas: ◦ ◦ ∂ (g (α + tδ)) = ( x , x + x , x )δk, ∂t ij ◦ h ik ji h i jki and ∂g 0 ik = ( x , x + x , x )(αm) . ∂s h im ki h i kmi 44 3. LOCAL GEOMETRY OF SURFACES

Proof. We have

∂ ∂ (g (α + tδ)) = x (α + tδ)), x (α + tδ) ∂t ij ◦ ∂th i ◦ j ◦ i ∂ ∂ = (x (α + tδ)), x + x , (x (α + tδ)) ∂t i ◦ j i ∂t j ◦     = x δk, x + x , x δk = ( x , x + x , x )δk. h ik ji h i jk i h ik ji h i jki 0 m 0 m 0 Furthermore, g = x , x 0 = x (α ) , x + x , x (α ) .  ik h i ki h im ki h i km i Lemma 3.8.2. Let f C0[a, b]. Suppose that for all g C0[a, b] b ∈ ∈ we have a 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. R b 2 Proof. We try the test function g(x) = f(x). Then a (f(x)) ds = 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 cRhoose g≡(x) = (x a)(b x)f(x).  − − Theorem 3.8.3. Suppose β = x α is a critical point of the energy functional (endpoints fixed). Then β◦satisfies the differential equation

( k) (αk)00 + Γk (αi)0(αj)0 = 0. ∀ ij Proof. We use Lemma 3.8.1 to evaluate the term A from equa- tion (3.8.2) as follows:

b 0 0 A = ( x , x + x , x )(αi) (αj) δkds h ik ji h i jki Za b 0 0 = 2 x , x αi αj δkds, h ik ji Za since summation is over both i and j. Similarly,

b 0 0 i j B = 2 gijα δ ds a Z 0 b 0 i j = 2 gijα δ ds − a Z   by integration by parts, where the boundary term vanishes since δ(a) = δ(b) = 0. Hence b i 0 k B = 2 gikα 0 δ ds − a Z   3.9. PRINCIPAL CURVATURES 45 by changing an index of summation. Thus

b 1 d i j i 0 k ( ) = xik, xj α 0α 0 gikα 0 δ ds. 2 dt E a h i − t=0 Z    

k Since this is true for any variation (δ ), by Lemma 3.8.2 we obtain the Euler-Lagrange equation

( k) x , x αi0αj0 g αi0 0 0, ∀ h ik ji − ik ≡ or   i j i i x , x α 0α 0 g 0α 0 g α 00 = 0. (3.8.3) h ik ji − ik − ik Using the formula from Lemma 3.8.1 for the s-derivative of the first fundamental form, we can rewrite the formula (3.8.3) as follows:

0 0 0 0 0 0 00 0 = x , x αi αj x , x αm αi x , x αi αm g αi h ik ji − h im ki − h i kmi − ik 0 0 00 = Γn x , x αm αi g αi −h im n ki − ik 0 0 00 = Γn g αm αi g αi , − im nk − ik where the cancellation of the first and the third term in the first line results from replacing index i by j and m by i in the third term. This is true k. Now multiply by gjk : ∀ 0 0 0 00 jk n m0 i jk i j n m0 i j i g Γimgnkα α + g gikα = δnΓimα α + δi α 0 j m0 i j 00 = Γimα α + α = 0, which is the desired geodesic equation. 

3.9. Principal curvatures

Definition 3.9.1. The principal curvatures, denoted k1 and k2, are the eigenvalues of the Weingarten map W .

Remark 3.9.2. The curvatures k1 and k2 are real. This follows from the selfadjointness of W (Theorem 3.6.12) and Corollary 2.2.7. Theorem 3.9.3. The Gaussian curvature equals the product of the principal curvatures. Proof. The determinant of a 2 by 2 matrix equals the product of i its eigenvalues: K = det(L j) = k1k2.  46 3. LOCAL GEOMETRY OF SURFACES

Theorem 3.9.4. Let v be a unit eigenvector belonging to a principal curvature k. Let β(s) be a geodesic satisfying β0(0) = v. Then the curvature of β as a space curve is the absolute value of k : k (0) = k . β | | i Proof. Let v = v xi, then W (v) = kv translates into i j i L jv = kv . (3.9.1) As before we may write β = x α. Meanwhile, by Theorem 3.7.7, we have ◦ kβ = II(β0, β0)  0 0 i j = Lijα α i j = Lijv v = Lm g vivj. − j mi Therefore from equation (3.9.1) we obtain k = k vmg vi  β − mi = k v 2 − k k = k, − proving the theorem.  Corollary 3.9.5. Gaussian curvature at a point p of a regular sur- face in R3 is the product of curvatures of two perpendicular geodesics passing through p, whose tangent vectors are eigenvectors of the Wein- garten map at the point. Definition 3.9.6. The Mean curvature H is half the trace of the 1 1 Weingarten map: H = (k + k ) = Li , cf. Table 4.3.1. 2 1 2 2 i 3.10. Minimal surfaces A surface x(u1, u2) is called minimal if H = 0 at every point, i.e. k1 + k2 = 0. Definition 3.10.1. A parametrisation x(u1, u2) is called isother- 1 2 2 mal if there is a function λ = λ(u , u ) such that gij = λ δij, i.e. x1, x1 = x , x and x , x = 0. h i h 2 2i h 1 2i Proposition 3.10.2. Assume x is isothermal. Then it satisfies the partial differential equation x + x = 2λ2Hn. 11 22 − 3.10. MINIMAL SURFACES 47

m m 2 2 i Proof. Note Lij = L jgmi = L jλ δmi = λ L j. Thus L 1 − L +−L − Li = ij and H = Li = 11 22 . Since x , x = 0, we j − λ2 2 i − 2λ2 h 1 2i ∂ have x , x = 0 x , x + x , x = 0. So x , x = ∂u2 h 1 2i ⇒ h 12 2i h 1 22i −h 12 2i x , x . Also x , x x , x = 0. So h 1 22i h 1 1i − h 2 2i ∂ ∂ 0 = x , x x , x ∂u1 h 1 1i − ∂u1 h 2 2i = 2 x , x 2 x , x h 11 1i − h 21 2i = 2 x , x + 2 x , x h 11 1i h 22 1i = 2 x + x , x . h 11 22 1i Similarly x + x , x = 0. Thus x + x is proportional to n. h 11 22 2i 11 22 Write x11 + x22 = cn. Then

c = x + x , n h 11 22 i = x , n + x , n h 11 i h 22 i = L11 + L22 = 2λ2H, − as required. 

Definition 3.10.3. Let f : R2 R, (u1, u2) f(u1, u2). The Laplacian, denoted ∆(f), is the second-order→ differen7→tial operator on functions, defined by

∂2f ∂2f ∆f = + . ∂(u1)2 ∂(u2)2

We say that f is harmonic if ∆f = 0.

Corollary 3.10.4. Let x(u1, u2) = (x(u1, u2), y(u1, u2), z(u1, u2)) in R3 be a parametrized surface and assume that x is isothermal. Then x is minimal if and only if the coordinate functions x, y, z are harmonic.

Proof. From Proposition 3.10.2 we have ∆(x) = 2λ2 H , prov- ing the corollary. k k | | 

Example 3.10.5. The catenoid is parametrized as follows:

x(θ, ϕ) = (a cosh ϕ cos θ, a cosh ϕ sin θ, aϕ). 48 3. LOCAL GEOMETRY OF SURFACES

2 2 Here f(ϕ) = a cosh ϕ, while g(ϕ) = aϕ. Then g11 = a cosh ϕ = 2 2 2 2 2 2 λ , g22 = (a sinh ϕ) + a = a cosh ϕ = λ , g12 = 0. Calculate: x + x = ( a cosh ϕ cos θ, a cosh ϕ sin θ, 0) 11 22 − − + (a cosh ϕ cos θ, a cosh ϕ sin θ, 0) = (0, 0, 0). Thus the catenoid is a minimal surface. In fact, it is the only surface of revolution which is minimal [We55, p. 179]. CHAPTER 4

The theorema egregium of Gauss

The present Chapter is an introduction to Gaussian curvature, and is thus simultaneously elementary, and essential to the book. It is true that some of the material in the book formally stands on its own even without understanding Gaussian curvature. Still, the material on CAT(0) metrics in genus 2 requires an understanding of negative Gaussian curvature, while Chapter on the filling area theorem requires an understanding of constant positive curvature metrics. To go beyond formalism, however, it would be very difficult to un- derstand the geometric classification of surfaces without understand- ing, first, the intrinsic nature of Gaussian curvature, i.e. the theorema egregium of Gauss. A beautiful historical account and an analysis of Gauss’s proof of the theorema egregium may be found in [Do79]. Our formula (4.3.3) for Gaussian curvature is actually very similar to the traditional formula for the Riemann curvature tensor in terms of the Levi-Civita connection (note the antisymmetrisation in both formulas, and the corresponding two summands), without the burden of the connection formalism. We present a compact formula for Gaussian curvature and its proof, along the lines of the argument in M. do Carmo’s book [Ca76]. The appeal of an old-fashioned, computational, coordinate notation proof is that it obviates the need for higher order objects such as connections, tensors, exponental map, etc, and is, hopefully, directly accessible to a student not yet familiar with the subject, cf. Remark 4.4.4. Remark 4.0.6. Throughout this Chapter, we will make use of the Einstein summation convention, i.e. suppress the summation sym- bol when the summation index occurs simultaneously as a subscript and a superscript in a formula, cf. (4.2.1). P 4.1. Intrinsic vs extrinsic properties We would like to distinguish two types of properties of a surface in Euclidean space: intrinsic and extrinsic. Understanding the extrin- sic/intrinsic dichotomy is equivalent to understanding the theorema 49 50 4. THE THEOREMA EGREGIUM OF GAUSS egregium of Gauss. The theorema egregium is the key insight lying at the foundation of differential geometry as conceived by B. Riemann in his essay [Ri1854] presented before the Royal Scientific Society of G¨ottingen a century and a half ago. The theorema egregium asserts that an infinitesimal invariant of a surface in Euclidean space, called Gaussian curvature, is an “intrinsic” invariant of the surface Σ. In other words, Gaussian curvature of Σ is independent of its isometric imbedding in Euclidean space. This theorem paves the way for an intrinsic definition of curvature in modern Riemannian geometry. We will prove that Gaussian curvature K is an intrinsic invariant of a surface in Euclidean space, in the following precise sense: K can be expressed in terms of the coefficients of the first fundamental form (see Section ) and their derivatives alone. A priori the possibility of thus expressing K is not obvious, as the naive definition of K involves the coefficients of the second fundamental form (alternatively, of the Weingarten map). Remark 4.1.1. The intrinsic nature of Gaussian curvature paves the way for a transition from classical differential geometry, to a more abstract approach of modern differential geometry. The distinction can be described roughly as follows. Classically, one studies surfaces in Eu- clidean space. Here the first fundamental form (gij) of the surface is the restriction of the Euclidean inner product. Meanwhile, abstractly, a surface comes equipped with a set of coefficients, which we deliberately denote by the same letters, (gij) in each coordinate patch, or equiva- lently, its element of length. One then proceeds to study its geometry without any reference to a Euclidean imbedding, cf. (5.1.2). Such an approach was pioneered in higher dimensions in Riemann’s essay. The essay contains a single formula [Ri1854, p. 292] (cf. [Sp79, p. 147]), namely the formula for the element of length of a surface of constant (Gaussian) curvature K α: ≡ 1 2 α 2 dx (4.1.1) 1 + 4 x qX (today, of course, we would incorpP orate a summation index as part of the notation), cf. formulas (5.1.3), (5.5.1), and (5.5.2).

4.2. Preliminaries to the theorema egregium Given a surface Σ in 3-space which is the graph of a function of two variables, consider a critical point p Σ. ∈ 4.2. PRELIMINARIES TO THE THEOREMA EGREGIUM 51

Definition 4.2.1. The Gaussian curvature of the surface at p is the determinant of the Hessian of the function, i.e. the determinant of the two-by-two matrix of its second derivatives. The implicit function theorem allows us to view any point of a regular surface, as such a critical point, after a suitable rotation. We have thus given the simplest possible definition of Gaussian curvature at any point of a regular surface. Remark 4.2.2. The appeal if this definition is that it allows one immediately to grasp the basic distinction between negative versus positive curvature, in terms of the dichotomy “saddle point versus cup/cap”. It will be more convenient to use an alternative definition in terms of the Weingarten map, which is readily shown to be equivalent to Definition 4.2.1. We denote the coordinates in R2 by (u1, u2). In R3, let , be the Euclidean inner product. Let x = x(u1, u2) : R2 R3 beh airegular parametrized surface. Here “regular” means that→the vector valued function x has differential dx of rank 2 at every point. Consider partial ∂ derivatives xi = ∂ui (x), where i = 1, 2. Thus, vectors x1 and x2 form a basis for the tangent plane at every point. Similarly, let ∂2x x = R3. ij ∂ui∂uj ∈ Let n = n(u1, u2) be a unit normal to the surface at the point x(u1, u2), so that n, x = 0. h ii k Definition 4.2.3 (symbols gij, Lij, Γij). The first fundamental form (g ) is given in coordinates by g = x , x . The second fun- ij ij h i ji damental form (Lij) is given in coordinates by Lij = n, xij . The k h i symbols Γij are uniquely defined by the decomposition k xij = Γijxk + Lijn 1 2 (4.2.1) = Γijx1 + Γijx2 + Lijn, where the repeated (upper and lower) index k implies summation, in accordance with the Einstein summation convention, cf. Remark 4.0.6. i i Definition 4.2.4 (symbols L j). The Weingarten map (L j) is an endomorphism of the tangent plane, namely Rx1 Rx2. It is uniquely defined by the decomposition ⊕ i nj = L jxi 1 2 = L jx1 + L jx2, 52 4. THE THEOREMA EGREGIUM OF GAUSS

∂ where nj = ∂uj (n). Definition 4.2.5. We will denote by square brackets [ ] the anti- symmetrisation over the pair of indices found in between the brackets, e.g. a = 1 (a a ). [ij] 2 ij − ji k Note that g[ij] = 0, L[ij] = 0, and Γ[ij] = 0. We will use the k k notation Γij;` for the `-th partial derivative of the symbol Γij. Proposition 4.2.6. We have the following relation

Γq + Γm Γq = L Lq i[j;k] i[j k]m − i[j k] for each set of indices i, j, k, q (with, as usual, an implied summation over the index m).

Proof. Let us calculate the tangential component, with respect to the basis x1, x2, n , of the third partial derivative xijk. Recall p { } ` that nk = L kxp and xjk = Γjkx` + Likn. Thus, we have

m (xij)k = Γij xm + Lijn k m m = Γ ij;kxm + Γij x
mk + Lijnk + Lij;kn m m p p = Γij;kxm + Γij Γmkxp + Lmkn + Lij (L kxp) + Lij;kn.   Grouping together the tangential terms, we obtain

m m p p (xij)k = Γij;kxm + Γij Γmkxp + Lij (L kxp) + (. . .)n

q m q  q = Γij;k + Γij Γmk + LijL k xq + (. . .)n

 q m q q  = Γij;k + Γij Γkm + LijL k xq + (. . .)n,

 q  since the symbols Γkm are symmetric in the two subscripts. Now the symmetry in j, k (equality of mixed partials) implies the following iden- tity: (xi[j)k] = 0. Therefore

0 = (xi[j)k]

q m q q = Γi[j;k] + Γi[jΓk]m + Li[jL k] xq + (. . .)n,   q m q q  and therefore Γi[j;k] + Γi[jΓk]m + Li[jL k] = 0 for each q = 1, 2. 4.3. THE THEOREMA EGREGIUM OF GAUSS 53

4.3. The theorema egregium of Gauss Definition 4.3.1. The Gaussian curvature function K = K(u1, u2) is the determinant of the Weingarten map: i 1 2 K = det(L j) = 2L [1L 2]. (4.3.1) Theorem 4.3.2. We have the following three equivalent formulas for Gaussian curvature K: i 1 2 (a) K = det(L j) = 2L [1L 2]; (b) K = det(Lij ) ; det(gij ) (c) K = 2 L L2 . − g11 1[1 2] Proof. The first formula is our definition of K. To prove the sec- k ond formula, we use the relation Lij = L jgki. By the multiplicativity of determinant with respect to matrix−multiplication, we have

i 2 det(Lij) det(L j) = ( 1) . − det(gij) Let us now prove formula (c). The proof is a calculation. Note that by definition, 2L L2 = L L2 L L2 . Hence 1[1 2] 11 2 − 12 1 2L L2 = L1 g L2 g L2 L1 g L2 g L2 − 1[1 2] 1 11 − 1 21 2 − 2 11 − 2 21 1 1 2 2 2 1 2 2 2 = g11L 1L 2 g21L
1L 2 g11L 2L 1 + g21L
2L 1 − − (4.3.2) = g L1 L2 L1 L2 11 1 2 − 2 1 i = g11det (L j) = g11K.
k Note that we can either calculate using the formula Lij = L jgki, k − or the formula Lij = L igkj. Only the former one leads to the appro- priate cancellations as−in (4.3.2).  Theorem 4.3.3 (Theorema egregium). The Gaussian curvature function K = K(u1, u2) can be expressed in terms of the coefficients of the first fundamental form alone (and their first and second derivatives) as follows: 2 2 j 2 K = Γ1[1;2] + Γ1[1Γ2]j , (4.3.3) g11 k   where the symbols Γij can be expressed in terms of the derivatives of gij be the formula Γk = 1 (g g + g )g`k, where (gij) is the inverse ij 2 i`;j − ij;` j`;i matrix of (gij).

Remark 4.3.4. Unlike Gaussian curvature K, the mean curva- 1 i ture H = 2 L i cannot be expressed in terms of the gij and their deriva- tives. Indeed, the plane and the cylinder have parametrisations with 54 4. THE THEOREMA EGREGIUM OF GAUSS

Weingarten Gaussian cur- mean cur- i map (L j) vature K vature H 0 0 plane 0 0 0 0   1 0 cylinder 0 1 0 0 2 invariance   yes no of curvature Table 4.3.1. Plane and cylinder have the same intrinsic geometry (K), but different extrinsic geometries (H)

identical gij, but with different mean curvature, cf. Table 4.3.1. To summarize, Gaussian curvature is an intrinsic invariant, while mean curvature an extrinsic invariant, of the surface. Proof of theorema egregium. We present a streamlined version of do Carmo’s proof [Ca76, p. 233]. The proof is in 3 steps.

(1) We express the third partial derivative xijk in terms of the Γ’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 combination of the L’s. (3) The combination of the L’s is expressed in terms of Gaussian curvature. The first two steps were carried out in Proposition 4.2.6. Choosing the valus i = j = 1 and k = q = 2 for the indices, we obtain Γ2 + Γm Γ2 = L L2 1[1;2] 1[1 2]m − 1[1 2] i 2 = g1iL [1L 2] 1 2 = g11L [1L 2] 2 2 since the term L [1L 2] = 0 vanishes. Applying Theorem 4.3.2(c), we obtain the desired formula for K and complete the proof of the theorema egregium. 

4.4. The Laplacian formula for Gaussian curvature An (x, y) chart in which the metric becomes conformal (see Def- inition 5.1.5 of Chapter 5) to the standard flat metric dx2 + dy2, is 4.4. THE LAPLACIAN FORMULA FOR GAUSSIAN CURVATURE 55

1 2 Γij j = 1 j = 2 Γij j = 1 j = 2 i = 1 µ1 µ2 i = 1 µ2 µ1 i = 2 µ µ i = 2 µ− µ 2 − 1 1 2 k 2µ(u1,u2) Table 4.4.1. Symbols Γij of a metric e δij referred to as isothermal coordinates. The existence of the latter is proved in [Bes87].

Definition 4.4.1. The Laplace-Beltrami operator for a metric λδij in isothermal coordinates is 1 ∂2 ∂2 ∆ = + . LB λ ∂(u1)2 ∂(u2)2   Corollary 4.4.2. Given a metric λ(x, y)(dx2 +dy2) in isothermal coordinates, its Gaussian curvature is minus half the Laplace-Beltrami operator of the log of the conformal factor λ: 1 K = ∆ log λ. (4.4.1) −2 LB Proof. Let λ = e2µ. We have from Table 4.4.1: 2Γ2 = Γ2 Γ2 = µ µ . 1[1;2] 11;2 − 12;1 − 22 − 11 It remains to verify that the ΓΓ term in the expression (4.3.3) for the Gaussian curvature vanishes: j 2 1 2 2 2 2Γ1[1Γ2]j = 2Γ1[1Γ2]1 + 2Γ1[1Γ2]2 = Γ1 Γ2 Γ1 Γ2 + Γ2 Γ2 Γ2 Γ2 11 21 − 12 11 11 22 − 12 12 = µ µ µ ( µ ) + ( µ )µ µ µ 1 1 − 2 − 2 − 2 2 − 1 1 = 0. Then from formula (4.3.3) we have 2 1 K = Γ2 = (µ + µ ) = ∆ µ. λ 1[1;2] −λ 11 22 − LB  Meanwhile, ∆LB log λ = ∆LB(2µ) = 2∆LB(µ), proving the result. Setting λ = f 2, we restate the corollary as follows. Corollary 4.4.3. Given a metric f 2(x, y)(dx2 + dy2) in isother- mal coordinates, its Gaussian curvature is minus the Laplace-Beltrami operator of the log of the conformal factor f: K = ∆ log f. (4.4.2) − LB 56 4. THE THEOREMA EGREGIUM OF GAUSS

This formula is exploited in Either one of the formulas (4.3.3), (4.4.1), or (4.4.2) can serve as the intrinsic definition of Gaussian cur- vature, replacing the extrinsic definition (4.3.1), cf. Remark 4.1.1. Remark 4.4.4. For a reader familiar with elements of Riemannian geometry, it is worth mentioning that the Jacobi equation

y00 + Ky = 0 of a Jacobi field y on M (expressing an infinitesimal variation by geodesics) sheds light on the nature of curvature in a way that no mere formula for K could. However, to prove the Jacobi equation, one would need to have already an intrinsically well-defined quantity on the left hand side, y00 + Ky, of the Jacobi equation. In particular, one would need an already intrinsic notion of curvature K. Thus, a proof of the theorema egregium necessarily precedes the deeper insights provided by the Jacobi equation. Similarly, the Gaussian curvature at p M is the first significant term in 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, re- quires introducing higher order notions such as the exponential map, which are usually understood at a later stage than the notion of Gauss- ian curvature, cf. [Ca76, Car92, Ch93, GaHL04]. CHAPTER 5

Global geometry of surfaces

5.1. Metric preliminaries Definition 5.1.1. By a (2-dimensional) closed Riemannian man- ifold we mean a compact subset Σ Rn whose every point has a (relative) open neighborhood diffeomorphic⊂ to R2. One can then use the diffeomorphism as a parametrisation, to cal- culate the coefficients gij (see Definition 4.2.3) of the first fundamental form, as, for example, in Theorem 5.3.4 and Example 3.1.9. The col- lection of all such data is then denoted by the pair (Σ, ), where is referred to as “the metric”. G G Remark 5.1.2. Differential geometers like the (Σ, ) notation, be- cause it helps separate the topology Σ from the geometryG . Strictly speaking, the notation is redundant, since the object alreadyG incorpo- rates all the information, including the topology. HoGwever, geometers have found it useful to use when one wants to emphasize the geom- etry, and Σ when one wantsGto emphasize the topology. Note that, as far as the intrinsic geometry of a Riemannian man- ifold is concerned, the imbedding in Rn referred to in Definition 5.1.1 is irrelevant to a certain extent, all the more so since certain basic examples, such as flat tori, are difficult to imbed in a transparent way. Example 5.1.3. The round 2-sphere S2 R3 defined by the equa- tion ⊂ 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 − − p 1 2 1 2 1 2 x1(u , u ) = (u , u , f(u , u )). Thus, each point of the open northern hemisphere admits a neigh- borhood diffeomorphic to a ball (and hence to R2). To cover the 1 2 1 2 1 2 southern hemisphere, use the chart x2(u , u ) = (u , u , f(u , u )). − 1 2 To cover the points on the equator, use in addition charts x3(u , u ) = 57 58 5. GLOBAL GEOMETRY OF SURFACES

1 1 2 2 1 2 1 1 2 2 1 2 (u , f(u , u ), u ), x4(u , u ) = (u , f(u , u ), u ), as well as the charts x5(u , u ) = 1 2 1 2 1 2 −1 2 1 2 (f(u , u ), u , u ), x6(u , u ) = (f(u , u ), u , u ). ∂ i Recall that we are working with dual bases ∂ui for vectors, and (du ) for covectors (i.e. elements of the dual space), such that
∂ dui = δi , (5.1.1) ∂uj j   i where δj is the Kronecker delta. We can thus write the first fundamen- tal form as follows: = g (u1, u2)duiduj (5.1.2) G ij with the Einstein summation convention. We will work with such data independently of any Euclidean imbedding, as discussed in Re- mark 4.1.1. For example, if the metric coefficients form an identity matrix, we obtain = du1 2 + du2 2 , (5.1.3) G where the interior superscript denotes
an index,
while exterior super- script denotes the squaring operation. Theorem 5.1.4. Define the area of (Σ, ) by means of the formula G 1 2 area = det(gij)du du , (5.1.4) U ZU X{ } q namely by choosing a partition U of Σ subordinate to a finite open cover as in Definition 5.1.1, performing{ } a separate integration in each open set, and summing the resulting areas. Then the total area is in- dependent of the partition. Proof. Consider a change from a coordinate chart (ui) to another coordinate chart, denoted (vα). In the overlap of the two domains, the coordinates can be expressed in terms of each other, e.g. v = v(u), and we have the 2 by 2 Jacobian matrix ∂(v1, v2) , ∂(u1, u2) which is the matrix of partial derivatives. Denote by ∂(v1, v2) Jac(u) = det . ∂(u1, u2)   It is shown in advanced calculus that for any function f(v) in a do- main D, one has

f(v)dv1dv2 = g(u)Jac(u)du1du2, (5.1.5) ZD ZD 5.1. METRIC PRELIMINARIES 59 where g(u) = f(v(u)). α Denote by g˜αβ the metric coefficients with respect to the chart (v ). Thus, in the case of a metric induced by a Euclidean imbedding defined by x = x(u) = x(u1, u2), we obtain a new parametrisation y(v) = x(u(v)). Then ∂y ∂y g˜ = , αβ ∂vα ∂vβ   ∂x ∂ui ∂x ∂uj = , ∂ui ∂vα ∂uj ∂vβ   ∂ui ∂uj ∂x ∂x = , ∂vα ∂vβ ∂ui ∂uj   ∂ui ∂uj = g . ij ∂vα ∂vβ Since determinant is multiplicative, we obtain ∂(u1, u2) 2 det(g˜ ) = det(g )det . αβ ij ∂(v1, v2)   Hence using equation (5.1.5), we can write 1 1 2 1 2 2 1 2 det (g˜αβ)dv dv = det (g˜αβ) Jac(u)du du 1 ∂(u1, u2) = det 2 (g ) det Jac(u)du1du2 ij ∂(v1, v2)   1 2 1 2 = det (gij) du du since inverse maps have reciprocal Jacobians by chain rule. Thus the integrand is unchanged and area is well defined.  i j i j Definition 5.1.5. Two metrics, = gijdu du and h = hijdu du , on Σ are called conformally equivalentG, or conformal for short, if there exists a function f = f(u1, u2) > 0 such that = f 2h, G in other words, g = f 2h i, j. (5.1.6) ij ij ∀ Definition 5.1.6. The function f above is called the conformal factor (note that sometimes it is more convenient to refer, instead, to the function λ = f 2 as the conformal factor). Remark 5.1.7. Note that the length of every vector at a point (u1, u2) 1 2 i ∂ is multiplied precisely by f(u , u ). More speficially, a vector v = v ∂ui which is a unit vector for the metric h, is “stretched” by a factor of f, 60 5. GLOBAL GEOMETRY OF SURFACES i.e. its length with respect to equals f. Indeed, the new length of v is G 1 ∂ ∂ 2 (v, v) = vi , vj G G ∂ui ∂uj   p 1 ∂ ∂ 2 = , vivj G ∂ui ∂uj     i j = gijv v 2 i j = pf hijv v i j = fp hijv v = f.p Definition 5.1.8. An equivalence class of metrics on Σ conformal to each other is called a conformal structure on Σ. Theorem 5.1.9 (Riemann mapping/uniformisation). Every metric on a connected surface is conformally equivalent to a metric of constant Gaussian curvature. From the complex analytic viewpoint, the uniformisation theorem states that every Riemann surface is covered by either the sphere, the plane, or the upper halfplane. Thus no notion of curvature is needed for the 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 uniformisation theorem.

5.2. Geodesic equation and closed geodesics The author has found that a number of mathematicians from fields other than differential geometry, while interested in systoles, need to be reminded what the definition of a geodesic is. Perhaps the simplest possible definition of a geodesic β on a surface in 3-space is in terms of the orthogonality of its second derivative β00 to the surface. The nonlinear second order ordinary differential equation defining a geodesic is, of course, the “true” if complicated definition. We will now prove the equivalence of the two definitions. Consider a plane curve R R2 s −α→ (u1,u2) 5.2. GEODESIC EQUATION AND CLOSED GEODESICS 61 where α = (α1(s), α2(s)). Let x : R2 R3 be a regular parametrisation of a surface in 3-space. Then the comp→ osition R R2 R3 s −α→ (u1,u2) −→x yields a curve β = x α. ◦ Definition 5.2.1. 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, 00 0 0 0 d (αk) + Γk (αi) (αj) = 0 where = , (5.2.1) ij ds meaning that d2αk dαi dαj ( k) + Γk = 0; ∀ ds2 ij ds ds (b) the vector β00 is perpendicular to the surface and one has i j β00 = Lijα 0α 0n. (5.2.2)

i To prove the equivalence, we write β = x α, then β0 = xiα 0 by chain rule. Furthermore, ◦

d i i j i k β00 = (x α)α 0 + x α 00 = x α 0α 0 + x α 00. ds i ◦ i ij k k Since xij = Γijxk + Lijn holds, we have

i j k k i j β00 L α 0α 0n = x α 00 + Γ α 0α 0 . − ij k ij Definition 5.2.2. A closed geodesic in a Riemannian 2-manifold Σ is defined equivalently as (1) a periodic curve β : R Σ satisfying the geodesic equa- → tion αk00 + Γk αi0αj0 = 0 in every chart x : R2 Σ, where, as ij → 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/L Σ satisfying the T → geodesic equation at each point, where LT = T Z R is the rank one lattice generated by T > 0. ⊂ Definition 5.2.3. The length L(β) of a path β : [a, b] Σ is calculated using the formula → b L(β) = β0(t) dt, k k Za 62 5. GLOBAL GEOMETRY OF SURFACES

i j i where v = gijv v whenever v = v xi. The energy is defined k k b 2 by E(β) = β0(t) dt. a kp k A closedR geodesic as in Definition 5.2.2, item 2 has length T . Remark 5.2.4. The geodesic equation (5.2.1) is the Euler-Lagrange equation of the first variation of arc length. Therefore when a path minimizes arc length among all neighboring paths connecting two fixed points, it must be a geodesic. A corresponding statement is valid for closed loops, cf. proof of Theorem 5.2.5. See also the introduction to Chapter 6. In Sections 5.3 and 5.4 we will give a complete description of the geodesics for the constant curvature sphere, as well as for flat tori. Theorem 5.2.5. Every free homotopy class of loops in a closed manifold contains a closed geodesic. Proof. We sketch a proof for the benefit of a curious reader, who can also check that the construction is independent of the choices in- volved. The relevant topological notions are defined in Section 5.6 and [Hat02]. A free homotopy class α of a manifold M corresponds to a conjugacy class g π (M). Pick an element g g . Thus g acts α ⊂ 1 ∈ α on the universal cover M˜ of M. Let fg : M R be the displacement function of g, i.e. → fg(x) = d(x,˜ g.x˜). Let x M be a minimum of f . A first variation argument shows that 0 ∈ g any length-minimizing path between x˜0 and g.x˜0 descends to a closed geodesic in M representing α, cf. [Car92, Ch93, GaHL04]. 

5.3. Surfaces of constant curvature By the uniformisation theorem 5.1.9, all surfaces fall into three types, according to whether they are conformally equivalent to metrics that 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 of its Euler characteristic, cf. formula (5.6.1). Theorem 5.3.1 (Constant positive curvature). There are only two surfaces, up to isometry, of constant Gaussian curvature K = +1. They are the round sphere S2 of Example 5.1.3; and the real projective plane, denoted RP2. 5.3. SURFACES OF CONSTANT CURVATURE 63

The latter can be defined as follows. Let m : S2 S2 denote the antipodal map of the sphere, i.e. the restriction of the→ map v v 3 7→ − 2 in R . Then m is an involution, i.e. each orbit of the Z2 action on S defined by f consists of a pair of antipodal points, p S2. On the set theoretic level, the real projective plane RP2 is the{set} of⊂ such orbits, 2 2 2 i.e. the quotient of S by the Z2 action. Denote by Q : S RP the quotient map. The smooth structure and 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 hij its metric coefficien◦ ts.− Then ∂y ∂y ∂x ∂x ∂x ∂x h = , = , = , = g . (5.3.1) ij h∂ui ∂uj i h−∂ui −∂uj i h∂ui ∂uj i ij Thus the opposite chart defines the identical metric coefficients. The 2 composition Q x is a chart on RP , and the same functions gij form ◦ 2 the metric coefficients for RP relative to this chart. We can summarize the preceeding discussion by means of the fol- lowing definition. Definition 5.3.2. The real projective plane RP2 is defined in the following two equivalent ways: (1) the quotient of the round sphere S2 by (the restriction to S2 of) the antipodal map v v in R3. In other words, a typical point of RP2 can be though7→ −t of as a pair of opposite points of the round sphere. (2) the northern hemisphere of S2, with opposite points of the equator identified. The smooth structure of RP2 is induced from the round sphere. Since the antipodal map preserves the metric coefficients by the calcu- lation (5.3.1), the metric structure of constant Gaussian curvature K = +1 descends to RP2, as well. Definition 5.3.3. A loop α : S1 X of a space X is called simple if the map α is one-to-one, cf. Definition→ 5.6.1. Theorem 5.3.4. All simple closed geodesics on S2 have length 2π. The simple closed geodesics on RP2 have length π. The closed geodesics in both cases are defined by the great circles on the sphere. Proof. We calculate the length of the equator of S2 parametrized by x(θ, φ) = (sin φ cos θ, sin φ sin θ, cos φ) in spherical coordinates (θ, φ). The equator is the curve x α where α(s) = (s, π ) with s [0, 2π]. ◦ 2 ∈ 64 5. GLOBAL GEOMETRY OF SURFACES

2 Recall that the metric coefficients are given by g11(θ, φ) = sin φ, while g22 = 1 and g12 = 0. Thus π i j π dθ 2 β0(s) = g s, α 0α 0 = sin = 1. k k ij 2 2 ds Thus the length of β is


2π 2π β0(s) ds = 1ds = 2π. k k Z0 Z0 A geodesic on RP2 is twice as short as on S2, since the antipodal points are identified, and therefore the geodesic “closes up” sooner than (i.e. twice as fast as) on the sphere. For example, a longitude of S2 is not a closed curve, but it descends to a closed curve on RP2, since its endpoints (north and south poles) are antipodal, and are therefore identified with each other. 

5.4. Flat surfaces 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(kL)k = dim(B⊂). Wkekdefine the notion of successive minima of L as follows, cf. [GruL87, p. 58]. Definition 5.4.1. For each k = 1, 2, . . . , rank(L), define the k-th successive minimum of the lattice L by

lin. indep. v1, . . . , vk L λk(L, ) = inf λ R ∃ ∈ . (5.4.1) k k ∈ with vi λ for all i  k k ≤ 

Thus the first successive minimum, λ (L, ) is the least length of 1 a nonzero vector in L. k k A metric is called flat if its Gaussian curvature K vanishes at ev- ery point. A closed surface of constant Gaussian curvature K = 0 is topologically either a torus T2 or a Klein bottle. Let us give a precise description in the former case. Example 5.4.2 (Flat tori). Every flat torus is isometric to a quo- tient 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 action of the lattice in R2. The smooth structure is inherited from R2. Meanwhile, the additive action of the lattice is isometric. Indeed, we have dist(p, q) = q p , while for any ` L, we have k − k ∈ dist(p + `, q + `) = q + ` (p + `) = q p = dist(p, q). k − k k − k Therefore the flat metric on R2 descends to T2. 5.5. HYPERBOLIC SURFACES 65

Note that locally, all flat tori are indistinguishable from the flat plane itself. However, their global geometry depends on the metric invariants of the lattice, e.g. its successive minima, cf. Definition 5.4.1. Thus, we have the following. Theorem 5.4.3. The least length of a nontrivial closed geodesic on 2 2 a flat torus T = R /L equals the first successive minimum λ1(L). Proof. The geodesics on the torus are the projections of straight lines in R2. In order for a straight line to close up, it must pass through a pair of points x and x+` where ` L. The length of the corresponding closed geodesic on T2 is precisely ∈` , where is the Euclidean norm. By choosing a shortest element kinkthe lattice,k k we obtain a shortest closed geodesic on the corresponding torus. 

5.5. Hyperbolic surfaces Most closed surfaces admit neither flat metrics nor metrics of pos- itive curvature, but rather hyperbolic metrics. A hyperbolic surface is a surface equipped with a metric of constant Gaussian curvature K = 1. This case is far richer than the other two. − Example 5.5.1. The pseudosphere (so called because its Gauss- ian curvature is constant, and equals 1) is the surface of revolu- tion (f(φ) cos θ, f(φ) sin θ, g(φ)) in R3 defined− by the functions f(ϕ) = ϕ 1/2 eϕ and g(ϕ) = 1 e2ψ dψ, where ϕ ranges through the in- 0 − Z 2 terval < ϕ 0. The usual
formulas g11 = f as well as g22 = 2 −∞ 2 ≤ df dg 2ϕ dϕ + dϕ yield in our case g11 = e , while     2 g = (eϕ) 2 + √1 e2ϕ 22 − = e2ϕ + 1  e2ϕ  − = 1. e2ϕ 0 Thus (g ) = . The pseudosphere has constant Gaussian ij 0 1   curvature 1, but it is not a closed surface (as it is unbounded in R3). − The metric 1 2 2 2 = (dx + dy ) (5.5.1) GH y2 in the upperhalf plane 2 = (x, y) y > 0 H { | } 66 5. GLOBAL GEOMETRY OF SURFACES has constant Gaussian curvature K = 1. In coordinates (u1, u2), we can write it, a bit awkwardly, as −

1 1 2 2 2 2 = du + du . GH (u2)2   2

The Riemannian manifold ( , 2 ) is referred to as the Poincar´e up- perhalf plane. Its significanceHis GthatH every closed hyperbolic surface Σ is isometric to the quotient of the Poincar´e upperhalf plane by the action of a suitable group Γ: Σ = 2/Γ. H Here the nonabelian group Γ is a discrete subgroup Γ P SL(2, R), a b ⊂ where a matrix A = acts on 2 = z C (z) > 0 by z c d H { ∈ |= } 7→ az+b   cz+d , cf. [Kato92] where the following theorem is proved. Theorem 5.5.2. Every geodesic in the Poincar´e upperhalf plane is either a vertical ray, or a semicircle perpendicular to the x-axis. Definition 5.5.3. A projection (R2, ) R to the u1-axis is called a Riemannian submersion if the metricGco→efficients in the horizontal direction are independent of u2, or more precisely, 2 2 = h(u1) du1 + k(u1, u2) du2 , (5.5.2) G for suitable functions h and k (note
that thematrix
of metric coeffi- cients is diagonal but not necessarily scalar). Being a Riemannian submersion is, of course, a local property, and we stated it for Euclidean space for convenience. We will use it mostly in the context of maps between compact manifolds. Example 5.5.4. The metric (5.5.1) of the Poincar´e upperhalf plane 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.

5.6. Topological preliminaries We would like to provide a self-contained explanation of the topo- logical ingredient which is necessary so as to understand the Loewner inequality, i.e. essentially the notion of a noncontractible loop and the fundamental group of a topological space X. See [Hat02, Chapter 1] for a more detailed account. Definition 5.6.1. A loop in X can be defined in one of two equiv- alent ways: 5.6. TOPOLOGICAL PRELIMINARIES 67

(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 5.6.2. The two definitions of a loop are equivalent. Proof. Consider the unique increasing linear function f : [a, b] 2π(t a) → − [0, 2π] which is one-to-one and onto. Thus, f(t) = b a . Given a map λ(eis) : S1 X, we associate to it a map β(t) = −λ eif(t) , and vice versa. → 
Definition 5.6.3. A loop S1 X is said to be contractible if the map of the circle can be extended→ to a continuous map of the unit disk D X, where S1 = ∂D. → Definition 5.6.4. A space X is called simply connected if every loop in X is contractible. Theorem 5.6.5. The sphere Sn Rn+1 which is the solution set 2 2 ⊂ 1 of x0 + . . . + xn = 1 is simply connected for n 2. The circle S is not simply connected. ≥ One can refine the notion of simple connectivity by introducing a group, denoted π1(X) = π1(X, x0), and called the fundamental group of X relative to a fixed “base” point x0 X. A based loop is a loop α : [0, 1] X satisfying∈ the condition α(0) = → α(1) = x0. In terms of the second item of Definition 5.6.1, we choose 1 i0 a fixed point s0 S . For example, we can choose s0 = e = 1 for the usual unit circle∈ S1 C, and require that α(s ) = x . Then the ⊂ 0 0 group π1(X) is the quotient of the space of all based loops modulo the equivalence equivalence relation defined by homotopies fixing the basepoint, cf. Definition 5.6.6. The equivalence class of the identity element is precisely the set of contractible loops based at x0. The equivalence relation can be described as follows for a pair of loops in terms of the second item of Definition 5.6.1. Definition 5.6.6. Two based loops α, β : S1 X are equivalent, 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 the cylinder, i.e. the basepoint does not move during the{ homotop} × y. Definition 5.6.7. An equivalence class of based loops is called a based homotopy class. Removing the basepoint restriction (as well as the constancy condition of Definition 5.6.6), we obtain a larger class called a free homotopy class (of loops). 68 5. GLOBAL GEOMETRY OF SURFACES

Composition of two loops is defined most conveniently in terms of item 1 of Definition 5.6.1, by concatenating their domains. Namely, the product of a pair of loops, α : [ 1, 0] X and β : [0, +1] X, is a loop α.β : [ 1, 1] X, which coincides− →with α and β in their→domains of definition.− The→product loop α.β is continuous since α(0) = β(0) = x0. Then Theorem 5.6.5 can be refined as follows. 1 n Theorem 5.6.8. We have π1(S ) = Z, while π1(S ) is the trivial group for all n 2. ≥ Definition 5.6.9. The 2-torus T2 is defined to be the following Cartesian product: T2 = S1 S1, and can thus be realized as a subset × T2 = S1 S1 R2 R2 = R4. × ⊂ × 2 2 2 We have π1(T ) = Z . The familiar doughnut picture realizes T as a subset in Euclidean space R3. Definition 5.6.10. A 2-dimensional closed Riemannian manifold (i.e. surface) Σ is called orientable if it can be realized by a subset of R3. Every imbedded closed surface in 3-space admits a continuous choice of a unit normal vector at every point. Note that no such choice is pos- sible for an imbedding of the Mobius band, see [Ar83] for more details on orientability and imbeddings. Theorem 5.6.11 (Gauss-Bonnet theorem). Every closed surface Σ satisfies the identity

1 2 1 2 K(u , u ) det(gij)du du = 2πχ(Σ), (5.6.1) Σ Z q where K is the Gaussian curvature function on Σ, whereas χ(Σ) is its Euler characteristic (here the integral is understood, as usual, in the sense of Theorem 5.1.4). See [Hat02] for a definition of the Euler characteristic. The latter is even for closed orientable surfaces, and the integer p = p(Σ) 0 defined by ≥ χ(Σ) = 2 2p − is called the genus of Σ. We have p(S2) = 0, while p(T2) = 1. In gen- eral, 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 (5.6.1) that the only compact orientable surface admit- ting flat metrics is the 2-torus. See [Ar83] for a friendly topological introduction to surfaces. CHAPTER 6

Inequalities of Loewner and Pu

6.1. Definition of systole The notions of contractible loops and simply connected spaces were reviewed in Section 5.6. We will denote by sysπ1( ), the infimum of lengths, referred to as the “systole” of , of a nonconG tractible loop β in a compact, non-simply-connected RiemannianG manifold (M, ): G

sysπ1( ) = inf length(β), (6.1.1) G β where the infimum is over all noncontractible loops β in M. In graph theory, a similar invariant is known as the girth [Tu47], see the dis- cussion in Subsection The relation of the notion of systole expressed by (6.1.1) to the systolic arrays of [Ku78] is yet to be clarified. See Section 3.8 for a further discussion of the term, by its promulgator. It can be shown that for a compact Riemannian manifold, the infi- mum is always attained, cf. Theorem 5.2.5. A loop realizing the mini- mum is necessarily a simple closed geodesic. In systolic questions about surfaces, integral-geometric identities play a particularly important role. Roughly speaking, there is an inte- gral identity relating area on the one hand, and an average of energies of a suitable family of loops, on the other. By the Cauchy-Schwarz inequality, there is an inequality relating energy and length squared, hence one obtains an inequality between area and the square of the systole. Such an approach works both for the Loewner inequality (6.4.1) and Pu’s inequality (6.2.2) (biographical notes on C. Loewner and P. Pu appear in respectively). One can prove an inequality for the M¨obius band this way, as well [Bl61b].

6.2. Isoperimetric inequality and Pu’s inequality Pu’s inequality can be thought of as an “opposite” isoperimetric inequality, 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 the 69 70 6. INEQUALITIES OF LOEWNER AND PU plane, and area A of the region bounded by the curve. Namely, every simple closed curve in the plane satisfies the inequality A L 2 . π ≤ 2π   This classical isoperimetric inequality is sharp, insofar as equality is attained only by a round circle. In the 1950’s, Charles Loewner’s student P. M. Pu [Pu52] proved the following theorem. Let RP2 be the real projective plane endowed with an arbitrary metric, i.e. an imbedding in some Rn. Then L 2 A , (6.2.1) π ≤ 2π   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 only for a metric of constant Gaussian curvature, namely antipodal quotient of a round sphere, cf. Section 5.3. In our systolic notation (6.1.1), Pu’s inequality takes the following form: sysπ ( )2 π area( ), (6.2.2) 1 G ≤ 2 G for every metric on RP2. See Theorem 5.3.4 for a discussion of the constant. The inequalitG y is proved in Section 7.5. Pu’s inequality can be generalized as follows. We will say that a surface is aspherical if it is not a 2-sphere. Theorem 6.2.1. Every aspherical surface (Σ, ) satisfies the opti- mal bound (6.2.2), attained precisely when, on theGone hand, the sur- face Σ is a real projective plane, and on the other, the metric is of constant Gaussian curvature. G The extension to aspherical surfaces follows from Gromov’s inequal- ity (6.2.3) below (by comparing the numerical values of the two con- stants). Namely, every aspherical compact surface (Σ, ) admits a metric ball G B = B 1 sysπ ( ) Σ p 2 1 G ⊂ of radius 1 sysπ ( ) which satisfies [Gro83, Corollary 5.2.B] 2 1 G
4 sysπ ( )2 area(B). (6.2.3) 1 G ≤ 3 We will need the following area bound in Chapter cf. [BuraZ80, Chapter 1, Theorem 5.3.1, p. 43]. 6.3. HERMITE AND BERGE-MAR´ TINET CONSTANTS 71

Proposition 6.2.2. Whenever a point x Σ lies on a two-sided loop which is minimizing in its free homotopy class,∈ cf. Definition 5.6.7, 1 the metric ball Bx(r) Σ of radius r 2 sysπ1( ) satisfies the esti- mate ⊂ ≤ G area (B (r)) 2r2. (6.2.4) x ≥ Proof. More generally, for an arbitrary 2-complex X, one can argue as follows. Let γ(s) be an arclength parametrisation of the loop, with γ(0) = x. If points γ(r) and γ( r) lie in distinct con- nected components of the r-level curve, then−it is easy to see that the element [γ] π1(X, γ(0)) splits off as a generator of an infinite cyclic factor in a ∈free product decomposition of the fundamental group. If X = Σ is a closed surface, its fundamental group does not admit such a decomposition. Hence γ(r) and γ( r) must lie in a common connected component, C Σ, of the level−curve. Let δ C be the two imbedded arcs connecting⊂ γ(r) to γ( r). An easy argumen ⊂ t shows − that length(δ+ δ ) 2r. We now perform an integration to deduce the bound (6.2.4)∪ −from≥ the coarea formula, see [KatzRS06] for more details.  A (different) generalisation of Pu’s inequality for the relative systole of a surface of negative Euler characteristic appears in Theorem 6.3. Hermite and Berg´e-Martinet constants

Let b N. The Hermite constant γb is defined in one of the following two equiv∈alent ways:

(1) γb is the square of the biggest first successive minimum, cf. Defi- nition 5.4.1, among all lattices of unit covolume; (2) γb is defined by the formula λ (L) √γ = sup 1 L (Rb, ) , (6.3.1) b vol(Rb/L)1/b ⊆ k k   b where the supremum is extended over all lattices L in R with a Euclidean norm . k k A lattice realizing the supremum is called a critical lattice. A crit- ical lattice may be thought of as the one realizing the densest packing Rb 1 in when we place balls of radius 2 λ1(L) at the points of L. The existence of the Hermite constant, as well as the existence of critical lattices, are both nontrivial results [Ca71]. Lemma 6.3.1. When b = 2, we have the following value for the Hermite constant: γ = 2 = 1.1547 . . .. The corresponding critical 2 √3 lattice is homothetic to the Z-span of the cube roots of unity in C. 72 6. INEQUALITIES OF LOEWNER AND PU

Proof. Consider a lattice L C = R2. Clearly, multiplying L by nonzero complex numbers does not⊂ change the value of the quotient λ (L)2 1 . area(C/L)

Choose a “shortest” vector z L, i.e. we have z = λ1(L). By re- 1 ∈ | | placing L by the lattice z− L, we may assume that the complex num- ber +1 C is a shortest element in the lattice. We will denote the new ∈ lattice by the same letter L, so that now λ1(L) = 1. Now complete the element +1 L to a Z-basis τ, +1 for L. Thus τ λ1(L) = 1. Con- sider the real∈ part (τ). We{can adjust} the basis| b|y≥adding a suitable < 1 1 integer to τ, so as to satisfy the condition 2 (τ) 2 . Then the second basis vector τ lies in the closure of−the≤standard< ≤ fundamental domain D = z C z > 1, (z) < 1 , (z) > 0 (6.3.2) ∈ | | |< | 2 = for the action of the group PSL(2, Z) in the upperhalf plane of C, see √3 [Ser73, p. 78]. The imaginary part satisfies (τ) 2 , with equality i π = ≥ i 2π possible in the following two cases: τ = e 3 or τ = e 3 . Moreover, if τ = r exp(iθ), then (τ) √3 sin θ = = . τ ≥ 2 τ | | | | Finally, we calculate the area of the parallelogram in C spanned by τ and +1, and write λ (L)2 1 2 1 = area(C/L) τ sin θ ≤ √3 | | to conclude the proof.  Example 6.3.2. In dimensions b 3, the Hermite constants are harder to compute, but explicit values≥(as well as the associated crit- 1 ical lattices) are known for small dimensions ( 8), e.g. γ = 2 3 = ≤ 3 1.2599 . . ., while γ4 = √2 = 1.4142 . . .. Note that γn is asymptotically linear in n, cf. (6.3.5).

A related constant γb0 is defined as follows, cf. [BeM].

Definition 6.3.3. The Berg´e-Martinet constant γb0 is defined by setting b γ0 = sup λ (L)λ (L∗) L (R , ) , (6.3.3) b 1 1 ⊆ k k b where the supremum is extended over all lattices L in R .

6.4. THE LOEWNER INEQUALITY 73

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 (5.1.1). h i

Thus, the constant γb0 is bounded above by the Hermite constant γb of (6.3.1). We have γ10 = 1, while for b 2 we have the following inequality: ≥ 2 γ0 γ b for all b 2. (6.3.4) b ≤ b ≤ 3 ≥ Moreover, one has the following asymptotic estimates: b b (1 + o(1)) γ0 (1 + o(1)) for b , (6.3.5) 2πe ≤ b ≤ πe → ∞ cf. [LaLS90, pp. 334, 337]. Note that the lower bound of (6.3.5) for the Hermite constant and the Berg´e-Martinet constant is nonconstruc- tive, but see [RT90] and [ConS99]. Definition 6.3.4. A lattice L realizing the supremum in (2.5.3) or (6.3.3) is called dual-critical. Rb Remark 6.3.5. The constants γb0 and the dual-critical lattices in are explicitly known for b 4, cf. [BeM, Proposition 2.13]. In partic- ular, we have γ = 1, γ =≤2 . 10 20 √3 Example 6.3.6. In dimension 3, the value of the Berg´e-Martinet 3 constant, γ30 = 2 = 1.2247 . . ., is slightly below the Hermite con- 1 stant γ3 = 2 3 =q1.2599 . . .. It is attained by the face-centered cu- bic lattice, which is not isodual [MilH73, p. 31], [BeM, Proposition 2.13(iii)], [CoS94]. 6.4. The Loewner inequality Historically, the first lower bound for the volume of a Riemannian manifold in terms of a systole is due to Charles Loewner. In 1949, Loewner proved the first systolic inequality, in a course on Riemannian geometry at Syracuse University, cf. [Pu52]. Namely, he showed the following result, whose proof appears in Section 7.2. Theorem 6.4.1 (C. Loewner). Every Riemannian metric on the torus T2 satisfies the inequality G sysπ ( )2 γ area( ), (6.4.1) 1 G ≤ 2 G where γ = 2 is the Hermite constant (6.3.1). A metric attain- 2 √3 ing the optimal bound (6.4.1) is necessarily flat, and is homothetic to the quotient of C by the lattice spanned by the cube roots of unity, cf. Lemma 6.3.1. 74 6. INEQUALITIES OF LOEWNER AND PU

In the next chapter, we will prove both the Loewner inequality, and Pu’s inequality. CHAPTER 7

Systolic applications of integral geometry

7.1. An integral-geometric identity

2 2 Let T be a torus with an arbitrary metric. Let T0 = R /L be the 2 flat torus conformally equivalent to T . Let `0 = `0(x) be a simple 2 closed geodesic of T0. Thus `0 is the projection of a line `˜0 R . Let `˜ R2 be the line parallel to `˜ at distance y > 0 from ` ⊂(here y ⊂ 0 0 we must “choose sides”, e.g. by orienting `˜0 and requiring `˜y to lie to ˜ the left of `0). Denote by `y T0 the closed geodesic loop defined by ˜ ⊂ the image of `y. Let y0 > 0 be the smallest number such that `y0 = `0, ˜ ˜ i.e. the lines `y0 and `0 both project to `0. 2 Note that the loops in the family `y T are not necessarily geodesics with respect to the metric of{ T}2. ⊂On the other hand, the family satisfies the following identity. Lemma 7.1.1 (An elementary integral-geometric identity). The met- ric on T2 satisfies the following identity: y0 2 area(T ) = E(`y)dy, (7.1.1) Z0 where E is the energy of a loop with respect to the metric of T2, see Definition 5.2.3. Proof. Denote by f 2 the conformal factor of T2 with respect to 2 the flat metric T0. Thus the metric on T can be written as f 2(x, y)(dx2 + dy2).

By Fubini’s theorem applied to a rectangle with sides lengthT0 (`0) and y0, combined with Remark 5.1.7, we obtain area(T2) = f 2dxdy T Z 0 y0 = f 2(x, y)dx dy Z0 Z`y ! y0 = E(`y)dy, Z0 75 76 7. SYSTOLIC APPLICATIONS OF INTEGRAL GEOMETRY proving the lemma.  Remark 7.1.2. The identity (7.1.1) can be thought of as the sim- plest integral-geometric identity, cf. equation (7.4.3).

7.2. 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, of the Loewner inequality (6.4.1) for the 2-torus, see also [CK03]. We present the following slight generalisation: there exists a pair of closed geodesics on (T2, ), of respective lengths λ and λ , such that G 1 2 λ λ γ area( ), (7.2.1) 1 2 ≤ 2 G and whose homotopy classes form a generating set for π (T2) = Z Z. 1 × Proof. The proof relies on the conformal representation φ : T (T2, ), 0 → G where T0 is flat, cf. uniformisation theorem 5.1.9. Here the map φ may 2 be chosen in such a way that (T , ) and T0 have the same area. Let f be the conformal factor, so that G = f 2 (du1)2 + (du2)2 G 1 2 2 2 as in formula (5.1.6), where (du ) + (du ) (lo
cally) is the flat metric. Let `0 be any closed geodesic in T0. Let `s be the family of { } 1 geodesics parallel to `0. Parametrize the family `s by a circle S of length σ, so that { } σ`0 = area(T0). Thus T S1 is a Riemannian submersion, cf. Definition 5.5.3. Then 0 → area(T2) = f 2. T Z 0 2 2 By Fubini’s theorem, we have area(T ) = 1 ds f dt. Therefore by S `s the Cauchy-Schwarz inequality, R R 2 fdt 2 `s 1 2 area(T ) ds = ds (length φ(`s)) . ≥ 1  `  ` 1 ZS R 0 0 ZS 2 σ 2 Hence there is an s0 such that area(T ) length φ(`s ) , so that ≥ `0 0 length φ(` ) ` . s0 ≤ 0 This reduces the proof to the flat case. Given a lattice in C, we choose a shortest lattice vector λ1, as well as a shortest one λ2 not proportional to λ1. The inequality now follows from Lemma 6.3.1. In the boundary 7.2. TWO PROOFS OF THE LOEWNER INEQUALITY 77 case of equality, one can exploit the equality in the Cauchy-Schwarz inequality to prove that the conformal factor must be constant. 

Alternative proof. Let `0 be any simple closed geodesic in T0. Since the desired inequality (7.2.1) is scale-invariant, we can assume that the loop has unit length:

lengthT0 (`0) = 1, and, moreover, that the corresponding covering transformation of the universal cover C = T˜0 is translation by the element 1 C. We complete 1 to a basis τ, 1 for the lattice of covering transformations∈ { } of T0. Note that the rectangle defined by z = x + iy C 0 < x < 1, 0 < y < (τ)] { ∈ | = } is a fundamental domain for T , so that area(T ) = (τ). The maps 0 0 = ` (x) = x + iy, x [0, 1] y ∈ parametrize the family of geodesics parallel to `0 on T0. Recall that the metric of the torus T is f 2(dx2 + dy2). Lemma 7.2.1. We have the following relation between the length and energy of a loop: length(` )2 E(` ). y ≤ y Proof. By the Cauchy-Schwarz inequality,

1 1 2 f 2(x, y)dx f(x, y)dx , ≥ Z0 Z0  proving the lemma. 

Now by Lemma 7.1.1 and Lemma 7.2.1, we have

(τ) = 2 area(T2) length(` ) dy. ≥ y Z0
Hence there is a y0 such that area(T2) (τ) length(` )2, ≥ = y0 so that length(` ) 1. This reduces the proof to the flat case. Given y0 ≤ a lattice in C, we choose a shortest lattice vector λ1, as well as a shortest one λ2 not proportional to λ1. The inequality now follows from Lemma 6.3.1.  78 7. SYSTOLIC APPLICATIONS OF INTEGRAL GEOMETRY

Remark 7.2.2. Define a conformal invariant called the capacity of an annulus as follows. Consider a right circular cylinder ζ = R/Z [0, κ] κ × based on a unit circle R/Z. Its capacity C(ζκ) is defined to be its height, C(ζκ) = κ. Recall that every annular region in the plane is con- formally equivalent to such a cylinder, and therefore we have defined a conformal invariant of an arbitrary annular region. Every annular 2 region R satisfies the inequality area(R) C(R) sysπ1(R) . Mean- while, if we cut a flat torus along a shortest≥loop, we obtain an annular 1 √3 region R with capacity at least C(R) γ2− = 2 . This provides an al- ternative proof of the Loewner theorem.≥ In fact, we have the following identity: confsys ( )2C( ) = 1, (7.2.2) 1 G G where confsys is the conformally invariant generalisation of the homol- ogy systole, while C( ) is the largest capacity of a cylinder obtained by cutting open the underlyingG conformal structure on the torus. Question 7.2.3. Is the Loewner inequality (6.4.1) satisfied by ev- ery orientable nonsimply connected compact surface? Inspite of its elementary nature, and considerable research devoted to the area, the question is still open. Recently the case of genus 2 was settled as well as genus g 20 ≥ 7.3. Hopf fibration and the Hamilton quaternions The circle action in C2 restricts to the unit sphere S3 C2, which therefore admits a fixed point free circle action. Namely, the⊂circle S1 = eiθ : θ R C acts on (z , z ) by { ∈ } ⊂ 1 2 iθ iθ iθ e .(z1, z2) = (e z1, e z2). 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 the complex projective line CP1). Moreover, the quotient map S3 S2, (7.3.1) → called the Hopf fibration, is a Riemannian submersion, for which the natural metric on the base is a metric of constant Gaussian curva- ture +4 rather than +1. The latter is explained by the fact that the 1 π maximal distance between a pair of S orbits is 2 rather than π, in view of the fact that a pair of antipodal points of S3 lies in a common orbit. 7.4. DOUBLE FIBRATION OF SO(3) AND INTEGRAL GEOMETRY ON S2 79

Furthermore, the sphere S3 can be thought of as the Lie group formed of the unit (Hamilton) quaternions in H = C2. The Lie group SO(3) can be identified with S3/ 1 . Therefore the fibration (7.3.1) de- scends to a fibration { } SO(3) S2, (7.3.2) → exploited in the next section. Remark 7.3.1. The group G = SO(3) can be identified with the unit tangent bundle of S2, as follows. Choose a fixed unit tangent vector v to S2, and send an element g G, acting on S2, to the image of v under dg : T S2 T S2. ∈ → Remark 7.3.2. Let us elaborate a bit further on the quaternionic viewpoint, with an eye to aplying it in Section 7.4. A choice of a purely imaginary Hamilton quaternion in H = R1+Ri+Rj +Rk, i.e. a linear combination of i, j, and k, specifies a complex structure on H. Such a complex structure leads to a fibration of the unit sphere, as in (7.3.1). 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 fibration is perpendicular to a fiber of the other fibration whenever two such fibers meet. The construction of two “perpendicular” fibrations also 3 descends to the quotient by the natural Z2 action on S .

7.4. 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 (6.2.2) in Sec- tion 7.5, as well as in our argument in Lemma The identity has its origin in results of P. Funk [Fu16] determining a symmetric function on the two-sphere from its great circle integrals; see [Hel99, Proposition 2.2, p. 59], as well as Preface therein. The sphere S2 is the homogeneous space of the Lie group SO(3), cf. (7.3.2), so that S2 = SO(3)/SO(2), cf. [Ar83, p. 82]. Denote 2 by SO(2)σ the fiber over (stabilizer of) a typical point σ S . The projection ∈ p : SO(3) S2 (7.4.1) → is a Riemannian submersion for the standard metric of constant sec- 1 tional curvature 4 on SO(3). The total space SO(3) admits another Riemannian submersion, which we denote q : SO(3) RP2, (7.4.2) → whose typical fiber ν is an orbit of the geodesic flow on SO(3) when g the latter is viewed as the unit tangent bundle of S2, cf. Remark 7.3.1. 80 7. SYSTOLIC APPLICATIONS OF INTEGRAL GEOMETRY

SO(2) RP2, dν G GG t9 GG q tt  GG tt GG tt g G# tt SO(3) : L uu: LL uu LLp uu LLL uu LL& uu & ν / (S2, dσ) great circle

Figure 7.4.1. Integral geometry on S2, cf. (7.4.1) and (7.4.2)

Here RP2 denotes the double cover of RP2, homeomorphic to the 2- sphere, cf. Definition 5.3.2. A fiber of fibration q is the collection of unit vectorsgtangent to a given directed closed geodesic (great circle) on S 2. 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 space of oriented great circles on the sphere, with measure dν. While fibration q may seem more “mysterious”gthan fibration p, the two are actually equivalent from the quaternionic point of view, cf. Remark 7.3.2. The diagram of Figure 7.4.1 illustrates the maps defined so far. Denote by E (ν) the energy, and by L (ν) the length, of a curve ν with respect to aG (possibly singular) metricG = f 2 . G G0 Theorem 7.4.1. We have the following integral-geometric identity: 1 area( ) = E (ν)dν. G 2π RPg2 G Z Proof. We apply Fubini’s theorem [Ru87, p. 164] twice, to both q and p, to obtain

E (ν)dν = dν f 2 p ν(t)dt RPg2 G RPg2 ◦ ◦ Z Z Zν  = f 2 p ◦ ZSO(3) (7.4.3) = f 2 dσ 1 2 ZS ZSO(2)σ  = 2π area( ), G completing the proof of the theorem.  7.6. A TABLE OF OPTIMAL SYSTOLIC RATIOS OF SURFACES 81

Proposition 7.4.2. Let f be a square integrable function on S2, which is positive and continuous except possibly for a finite number of 1 points where f either vanishes or has a singularity of type √r , Then there is a great circle ν such that the following inequality is satisfied: 2 f(ν(t))dt π f 2(σ)dσ, (7.4.4) ≤ 2 Zν  ZS where t is the arclength parameter, and dσ is the Riemannian measure of the metric 0 of the standard unit sphere. In other words, there is a G 2 2 great circle of length L in the metric f 0, so that L πA, where A G 2 ≤ is the Riemannian surface area of the metric f 0. In the boundary case of equality in (7.4.4), the function f must beGconstant. Proof. The proof is an averaging argument and shows that the average length of great circles is short. Comparing length and energy yields the inequality

2 RPg2 L (ν) G E (ν)dν. (7.4.5)  2π  ≤ RPg2 G R Z The formula now follows from Theorem 7.4.1, since area(RP 2) = 4π. In the boundary case of equality in (7.4.4), one must have equality also in inequality (7.4.5) relating length and energy. It follogws that the conformal factor f must be constant along every great circle. Hence f is constant everywhere on S2.  7.5. Proof of Pu’s inequality A metric on RP2 lifts to a centrally symmetric metric ˜ on S2. Applying PropGosition 7.4.2 to ˜, we obtain a great circle of ˜-lengthG L satisfying L2 πA, where AGis the Riemannian surface Garea of ˜. ≤L 2 π A L GA Thus we have 2 2 2 , where sysπ1( ) 2 while area( ) = 2 , proving inequality (6.2.2)≤ . See [Iv02] for anG alternativ≤ e proof.G

7.6. A table of optimal systolic ratios of surfaces Denote by SR(Σ) the supremum of the systolic ratios, sysπ ( )2 SR(Σ) = sup 1 G , area( ) G G ranging over all metrics on a surface Σ. The known values of the optimal systolic ratio areGtabulated in Figure 7.6.1. It is interesting to note that the optimal ratio for the Klein bottle RP2#RP2 is achieved by a singular metric, described in the references listed in the table. 82 7. SYSTOLIC APPLICATIONS OF INTEGRAL GEOMETRY

SR(Σ) numerical value where to find it π Σ = RP2 = [Pu52] 1.5707 Section 7.5 2 ≈ 4 infinite π (Σ) < [Gro83] < 1.3333 . . . (6.2.3) 1 3 Σ = T2 = 2 (Loewner) 1.1547 (6.4.1) √3 ≈ π RP2 RP2 Bav86, Bav06, Σ = # = 3/2 1.1107 [ 2 ≈ Sak88]

1 √ Σ of genus 2 > 3 2 + 1 > 0.8047 ? Σ of genus 3 8
> 0.6598 [Cal96] ≥ 7√3

Figure 7.6.1. Values for optimal systolic ratio SR of surface Σ CHAPTER 8

A primer on surfaces

In this Chapter, we collect some classical facts on Riemann surfaces. More specifically, we deal with hyperelliptic surfaces, real surfaces, and Katok’s optimal bound for the entropy of a surface. 8.1. Hyperelliptic involution Let Σ be an orientable closed Riemann surface which is not a sphere. By a Riemann surface, we mean a surface equipped with a fixed confor- mal structure, cf. Definition 5.1.8, while all maps are angle-preserving. Definition 8.1.1. A hyperelliptic involution of a Riemann sur- face Σ of genus p is a holomorphic (conformal) map, J : Σ Σ, satisfying J 2 = 1, with 2p + 2 fixed points. → Definition 8.1.2. A surface Σ admitting a hyperelliptic involution will be called a hyperelliptic surface. Remark 8.1.3. The involution J can be identified with the non- trivial element in the center of the (finite) automorphism group of Σ (cf. [FK92, p. 108]) when it exists, and then such a J is unique, cf. [Mi95, p.204]. It is known that the quotient of Σ by the involution J produces a conformal branched 2-fold covering Q : Σ S2 (8.1.1) → of the sphere S2. Definition 8.1.4. The 2p + 2 fixed points of J are called Weier- strass points. Their images in S2 under the ramified double cover Q of formula (8.1.1) will be referred to as ramification points. A notion of a Weierstrass point exists on any Riemann surface, but will only be used in the present text in the hyperelliptic case. Example 8.1.5. In the case p = 2, topologically the situation can be described as follows. A simple way of representing the figure 8 contour in the (x, y) plane is by the reducible curve (((x 1)2 + y2) 1)(((x + 1)2 + y2) 1) = 0 (8.1.2) − − − 83 84 8. A PRIMER ON SURFACES

(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 (8.1.2)−as a subset of R3. The boundary 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 to the sphere.

A similar example can be repackaged in a metrically more precise way as follows.

Example 8.1.6. We start with a round metric on RP2. Now attach a small handle. The orientable double cover Σ of the resulting surface can be thought of as the unit sphere in R3, with two little handles at- tached at north and south poles, i.e. at the two points where the sphere meets the z-axis. Then one can think of the hyperelliptic involution J as the rotation of Σ by π around the z-axis. The six fixed points are the six points of intersection of Σ with the z-axis. Furthermore, there is an orientation reversing involution τ on Σ, given by the restriction to Σ of the antipodal map in R3. The composition τ J is the reflection fixing the xy-plane, in view of the following matrix ◦identity: 1 0 0 1 0 0 1 0 0 −0 1 0 −0 1 0 = 0 1 0 . (8.1.3)  0 −0 1  0 −0 1 0 0 1 − −       2 Meanwhile, the induced orientation reversing involution τ0 on S can just as well be thought of as the reflection in the xy-plane. This is because, at the level of the 2-sphere, it is “the same thing as” the composition τ J . Thus the fixed circle of τ0 is precisely the equator, cf. formula (8.3.3).◦ Then one gets a quotient metric on S2 which is roughly that of the western hemisphere, with the boundary longitude folded in two, The metric has little bulges along the z-axis at north and south poles, which are leftovers of the small handle.

8.2. 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 Σ is defined by a suitable equation of the form

w2 = f(z) (8.2.1) 8.3. OVALLESS SURFACES 85 in C2, where f is a polynomial. On such an affine part, the map J is given by J(z, w) = (z, w), while the hyperelliptic quotient map Q : Σ S2 is represented b−y the projection onto the z-coordinate in C2. →A slight technical problem here is that the map Σ CP2, (8.2.2) → whose image is the compactification of the solution set of (8.2.1), is not an imbedding. Indeed, there is only one point at infinity, given in homogeneous coordinates by [0 : w : 0]. This point is a singularity. A way of desingularizing it using an explicit change of coordinates at infinity is presented in [Mi95, p. 60-61]. The resulting smooth surface is unique [DaS98, Theorem, p. 100]. Remark 8.2.1. To explain what happens “geometrically”, note that there are two points on our affine surface “above infinity”. This means that for a large circle z = r, there are two circles above it satis- fying equation (8.2.1) where |f |has even degree 2p +2 (for a Weierstrass point we would only have one circle). To see this, consider z = reia. As the argument a varies from 0 to 2π, the argument of f(z) will change by (2p + 2)2π. Thus, if (reia, w(a)) represents a continuous curve on our surface, then the argument of w changes by (2p +2)π, and hence we end up where we started, and not at w (as would be the case were the polynomial of odd degree). Thus there− are two circles on the surface over the circle z = r. We conclude that to obtain a smooth compact surface, we will|need| to add two points at infinity, cf. discussion around [FK92, formula (7.4.1), p. 102]. Thus, the affine part of Σ, defined by equation (8.2.1), is a Rie- mann surface with a pair of punctures p1 and p2. A neighborhood of each pi is conformally equivalent to a punctured disk. By replacing each punctured disk by a full one, we obtain the desired compact Rie- mann surface Σ. The point at infinity [0 : w : 0] CP2 is the image of ∈ both pi under the map (8.2.2). 8.3. Ovalless surfaces Denote by Σι the fixed point set of an involution ι of a Riemann surface Σ. Let Σ be a hyperelliptic surface of even genus p. Let J : Σ Σ be the hyperelliptic involution, cf. Definition 8.1.1. Let τ : Σ →Σ be a fixed point free, antiholomorphic involution. → Lemma 8.3.1. The involution τ commutes with J and descends 2 2 2 to S . The induced involution τ0 : S S is an inversion in a τ J → circle C0 = Q(Σ ◦ ). The set of ramification points is invariant under 2 the action of τ0 on S . 86 8. A PRIMER ON SURFACES

Proof. By the uniqueness of J, cf. Remark 8.1.3, we have the commutation relation τ J = J τ, (8.3.1) ◦ ◦ cf. relation (8.1.3). Therefore τ descends to an involution τ0 on the sphere. There are two possibilities, namely, τ is conjugate, in the group of fractional linear transformations, either to the map z z¯, or to the map z 1 . In the latter case, τ is conjugate to the an7→tipodal map 7→ − z¯ 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 a connected loop. Thus we get an action of Z2 Z2 on a loop, resulting in a contradiction. × In more detail, the set of the 2p + 2 ramification points on S2 is centrally symmetric. Since there is an odd number, p + 1, of ramifi- cation points in a hemisphere, a generic great circle A S2 has the 1 ⊂ property that its inverse image Q− (A) Σ is connected. Thus both involutions τ and J , as well as τ J , ⊂act fixed point freely on the 1 ◦ loop Q− (A) Σ, which is impossible. Therefore τ must fix a point. ⊂ 0 It follows that τ0 is an inversion in a circle.  Suppose a hyperelliptic Riemann surface Σ admits an antiholomor- phic involution τ. In the literature, the components of the fixed point set Στ of τ are sometimes referred to as “ovals”. When τ is fixed point free, we introduce the following terminology. Definition 8.3.2. A hyperelliptic surface (Σ, J) of even positive genus p > 0 is called ovalless real if one of the following equivalent conditions is satisfied: (1) Σ admits an imaginary reflection, i.e. a fixed point free, anti- holomorphic involution τ; (2) the affine part of Σ is the locus in C2 of the equation w2 = P (z), (8.3.2) − where P is a monic polynomial, of degree 2p + 2, with real coefficients, no real roots, and with distinct roots. Lemma 8.3.3. The two ovalless reality conditions of Definition 8.3.2 are equivalent. Proof. A related result appears in [GroH81, p. 170, Proposi- tion 6.1(2)]. To prove the implication (2) = (1), note that complex conjugation leaves the equation invariant, and⇒ therefore it also leaves invariant the locus of (8.3.2). A fixed point must be real, but P is positive hence (8.3.2) has no real solutions. There is no real solution at 8.4. KATOK’S ENTROPY INEQUALITY 87 infinity, either, as there are two points at infinity which are not Weier- strass points, since P is of even degree, as discussed in Remark 8.2.1. The desired imaginary reflection τ switches the two points at infinity, and, on the affine part of the Riemann surface, coincides with complex conjugation (z, w) (z¯, w¯) in C2. To prove the implication7→ (1) = (2), note that by Lemma 8.3.1, 2 ⇒ the induced involution τ0 on S = Σ/ J may be thought of as complex conjugation, by choosing the fixed circle of τ0 to be the circle R C = S2. (8.3.3) ∪ {∞} ⊂ ∪ {∞} Since the surface is hyperelliptic, it is the smooth completion of the locus in C2 of some equation of the form (8.3.2), cf. (8.2.1). Here P is 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 Weierstrass points. Hence the set of roots must be invariant under τ0. Thus the roots of the polynomial either come in conjugate pairs, or else are real. Therefore P has real coefficients. Furthermore, the leading cofficient of P may be absorbed into the w-coordinate by extracting a square root. Here we may have to rotate w by i, but at any rate the coefficients of 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, as well! This contradicts the fixed point freeness of τ. Thus all roots of P must come in conjugate pairs. 

8.4. Katok’s entropy inequality Let (Σ, ) be a closed surface with a Riemannian metric. Denote G by (Σ˜, ˜) the universal Riemannian cover of (Σ, ). Choose a point x˜ G G 0 ∈ Σ.˜ Definition 8.4.1. The volume entropy (or asymptotic volume) h(M, ) of a surface (Σ, ) is defined by setting G G log vol ˜ B(x˜0, R) h(Σ, ) = lim G , (8.4.1) G R + R → ∞
where vol ˜ B(x˜0, R) is the volume (area) of the ball of radius R centered G at x˜ Σ˜. 0 ∈ Since Σ is compact, the limit in (8.4.1) exists, and does not depend on the point x˜0 Σ˜ [Ma79]. This asymptotic invariant describes the exponential growth∈ rate of the volume in the universal cover. 88 8. A PRIMER ON SURFACES

Definition 8.4.2. The minimal volume entropy, MinEnt, of Σ is the infimum of the volume entropy of metrics of unit volume on Σ, or equivalently 1 MinEnt(Σ) = inf h(Σ, ) vol(Σ, ) 2 (8.4.2) G G G where runs over the space of all metrics on Σ. For an n-dimensional manifoldG in place of Σ, one defines MinEnt similarly, by replacing the 1 exponent of vol by n . The classical result of A. Katok [Kato83] states that every metric on a closed surface Σ with negative Euler characteristic χ(Σ) satisfiesG the optimal inequality 2π χ(Σ) h( )2 | |. (8.4.3) G ≥ area( ) G Inequality (8.4.3) also holds for hom ent( ) [Kato83], as well as the topological entropy, since the volume entropG y bounds from below the topological entropy (see [Ma79]). We recall the following well-known fact, cf. [KatH95, Proposition 9.6.6, p. 374].

Lemma 8.4.3. Let (M, ) be a closed Riemannian manifold. Then, G log(P (T )) h(M, ) = lim 0 (8.4.4) T + G → ∞ T where P 0(T ) is the number of homotopy classes of loops based at some fixed point x0 which can be represented by loops of length at most T . Proof. Let x M and choose a lift x˜ M˜ . The group 0 ∈ 0 ∈

Γ := π1(M, x0) acts on M˜ by isometries. The orbit of x˜0 under Γ is denoted Γ.x˜0. Consider a fundamental domain ∆ for the action of Γ, containing x˜0. Denote by D the diameter of ∆. The cardinal of Γ.x˜0 B(x˜0, R) is bounded from above by the number of translated fundamen∩ tal do- mains γ.∆, where γ Γ, contained in B(x˜0, R + D). It is also bounded from below by the ∈number of translated fundamental domains γ.∆ contained in B(x˜0, R). Therefore, we have vol(B(x˜ , R)) vol(B(x˜ , R + D)) 0 card (Γ.x˜ B(x˜ , R)) 0 . vol(M, ) ≤ 0 ∩ 0 ≤ vol(M, ) G G (8.4.5) 8.4. KATOK’S ENTROPY INEQUALITY 89

Take the log of these terms and divide by R. The lower bound becomes 1 vol(B(R)) log = R vol( )  G  (8.4.6) 1 1 = log (vol(B(R))) log (vol( )) , R − R G and the upper bound becomes 1 vol(B(R + D)) log = R vol( )  G  (8.4.7) R + D 1 1 = log(vol(B(R + D))) log(vol( )). R R + D − R G Hence both bounds tend to h( ) when R goes to infinity. Therefore, G 1 h( ) = lim log (card(Γ.x˜0 B(x˜0, R))) . (8.4.8) R + G → ∞ R ∩ This yields the result since P 0(R) = card(Γ.x˜ B(x˜ , R)).  0 ∩ 0

Bibliography

[Ab81] Abikoff, W.: The uniformization theorem. Amer. Math. Monthly 88 (1981), no. 8, 574–592. [Ac60] Accola, R.: Differentials and extremal length on Riemann surfaces. Proc. Nat. Acad. Sci. USA 46 (1960), 540–543. [Am04] Ammann, B.: Dirac eigenvalue estimates on two-tori. J. Geom. Phys. 51 (2004), no. 3, 372–386. [Ar83] Armstrong, M.: Basic topology. Corrected reprint of the 1979 original. Undergraduate Texts in Math. Springer-Verlag, New York-Berlin, 1983. [Ar74] Arnold, V.: Mathematical methods of classical mechanics. Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein. Cor- rected reprint of the second (1989) edition. Graduate Texts in Mathe- matics, 60. Springer-Verlag, New York, 199?. [Ba93] Babenko, I.: Asymptotic invariants of smooth manifolds. Russian Acad. Sci. Izv. Math. 41 (1993), 1–38. [Ba02a] Babenko, I. Forte souplesse intersystolique de vari´et´es ferm´ees et de poly`edres. Annales de l’Institut Fourier 52, 4 (2002), 1259-1284. [Ba02b] Babenko, I. Loewner’s conjecture, Besicovich’s example, and relative systolic geometry. [Russian]. Mat. Sbornik 193 (2002), 3-16. [Ba04] Babenko, I.: G´eom´etrie systolique des vari´et´es de groupe fondamen- tal Z2, S´emin. Th´eor. Spectr. G´eom. Grenoble 22 (2004), 25-52. [BB05] Babenko, I.; Balacheff, F.: G´eom´etrie systolique des sommes connexes et des revˆetements cycliques, Mathematische Annalen (to appear). [BaK98] Babenko, I.; Katz, M.: Systolic freedom of orientable manifolds, Annales Scientifiques de l’E.N.S. (Paris) 31 (1998), 787-809. [Bana93] Banaszczyk, W. New bounds in some transference theorems in the ge- ometry of numbers. Math. Ann. 296 (1993), 625–635. [Bana96] Banaszczyk, W. Inequalities for convex bodies and polar reciprocal lat- tices in Rn II: Application of K-convexity. Discrete Comput. Geom. 16 (1996), 305–311. [BCIK04] Bangert, V; Croke, C.; Ivanov, S.; Katz, M.: Boundary case of equality in optimal Loewner-type inequalities, Trans. Amer. Math. Soc., to appear. See arXiv:math.DG/0406008 [BCIK05] Bangert, V; Croke, C.; Ivanov, S.; Katz, M.: Filling area conjecture and ovalless real hyperelliptic surfaces, Geometric and Functional Analysis (GAFA) 15 (2005) no. 3. See arXiv:math.DG/0405583 [BK03] Bangert, V.; Katz, M.: Stable systolic inequalities and cohomology products, Comm. Pure Appl. Math. 56 (2003), 979–997. Available at arXiv:math.DG/0204181

91 92 BIBLIOGRAPHY

[BK04] Bangert, V; Katz, M.: An optimal Loewner-type systolic inequality and harmonic one-forms of constant norm. Comm. Anal. Geom. 12 (2004), number 3, 703-732. See arXiv:math.DG/0304494 [Bar57] Barnes, E. S.: On a theorem of Voronoi. Proc. Cambridge Philos. Soc. 53 (1957), 537–539. [Bav86] Bavard, C.: In´egalit´e isosystolique pour la bouteille de Klein. Math. Ann. 274 (1986), no. 3, 439–441. [Bav88] Bavard, C.: In´egalit´es isosystoliques conformes pour la bouteille de Klein. Geom. Dedicata 27 (1988), 349–355. [Bav92] Bavard, C.: La systole des surfaces hyperelliptiques, Prepubl. Ec. Norm. Sup. Lyon 71 (1992). [Bav06] Bavard, C.: Une remarque sur la g´eom´etrie systolique de la bouteille de Klein. Arch. Math. (Basel) 87 (2006), no. 1, 72–74. [BeM] Berg´e, A.-M.; Martinet, J. Sur un probl`eme de dualit´e li´e aux sph`eres en g´eom´etrie des nombres. J. Number Theory 32 (1989), 14–42. [Ber72] Berger, M. A l’ombre de Loewner. Ann. Scient. Ec. Norm. Sup. (S´er. 4) 5 (1972), 241–260. [Ber80] Berger, M.: Une borne inf´erieure pour le volume d’une vari´et´e rieman- nienne en fonction du rayon d’injectivit´e. Ann. Inst. Fourier 30 (1980), 259–265. [Ber93] Berger, M. Systoles et applications selon Gromov. S´eminaire N. Bour- baki, expos´e 771, Ast´erisque 216 (1993), 279–310. [Ber76] Berstein, I.: On the Lusternik–Schnirelmann category of Grassmannians, Proc. Camb. Phil. Soc. 79 (1976), 129-134. [Bes87] Besse, A.: Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 10. Springer-Verlag, Berlin, 1987. [BesCG95] Besson, G.; Courtois, G.;, Gallot, S.: Volume et entropie minimale des espaces localement sym´etriques. Invent. Math. 103 (1991), 417–445. [BesCG95] Besson, G.; Courtois, G.;, Gallot, S.: Hyperbolic manifolds, amalga- mated products and critical exponents. C. R. Math. Acad. Sci. Paris 336 (2003), no. 3, 257–261. [Bl61] Blatter, C. Ub¨ er Extremall¨angen auf geschlossenen Fl¨achen. Comment. Math. Helv. 35 (1961), 153–168. [Bl61b] Blatter, C.: Zur Riemannschen Geometrie im Grossen auf dem M¨obius- band. Compositio Math. 15 (1961), 88–107. [Bo94] Borzellino, J. E.: Pinching theorems for teardrops and footballs of rev- olution. Bull. Austral. Math. Soc. 49 (1994), no. 3, 353–364. [BG76] Bousfield, A.; Gugenheim, V.: On PL de Rham theory and rational homotopy type. Mem. Amer. Math. Soc. 8, 179, Providence, R.I., 1976. [BI94] Burago, D.; Ivanov, S.: Riemannian tori without conjugate points are flat. Geom. Funct. Anal. 4 (1994), no. 3, 259–269. [BI95] Burago, D.; Ivanov, S.: On asymptotic volume of tori. Geom. Funct. Anal. 5 (1995), no. 5, 800–808. [BuraZ80] Burago, Y.; Zalgaller, V.: Geometric inequalities. Grundlehren der Mathematischen Wissenschaften 285. Springer Series in Soviet Math- ematics. Springer-Verlag, Berlin, 1988 (original Russian edition: 1980). BIBLIOGRAPHY 93

[BS94] Buser, P.; Sarnak, P.: On the period matrix of a Riemann surface of large genus. With an appendix by J. H. Conway and N. J. A. Sloane. Invent. Math. 117 (1994), no. 1, 27–56. [Cal96] Calabi, E.: Extremal isosystolic metrics for compact surfaces, Actes de la table ronde de geometrie differentielle, Sem. Congr. 1 (1996), Soc. Math. France, 167–204. [Ca76] Manfredo do Carmo, Differential geometry of curves and surfaces. Prentice-Hall, 1976. [Car92] do Carmo, M.: Riemannian geometry. Translated from the second Por- tuguese edition by Francis Flaherty. Mathematics: Theory & Applica- tions. Birkh¨auser Boston, Inc., Boston, MA, 1992. [Ca71] Cassels, J. W. S.: An introduction to the geometry of numbers. Sec- ond printing. Grundlehren der mathematischen Wissenschaften, 99. Springer-Verlag, Berlin-Heidelberg-New York, 1971. [Ch93] Chavel, I.: Riemannian geometry—a modern introduction. Cambridge Tracts in Mathematics, 108. Cambridge University Press, Cambridge, 1993. [CoS94] Conway, J.; Sloane, N.: On lattices equivalent to their duals. J. Number Theory 48 (1994), no. 3, 373–382. [ConS99] Conway, J. H.; Sloane, N. J. A.: Sphere packings, lattices and groups. Third edition. Springer, 1999. [CoT03] Cornea, O.; Lupton, G.; Oprea, J.; Tanr´e, D.: Lusternik-Schnirelmann category. Mathematical Surveys and Monographs, 103. American Math- ematical Society, Providence, RI, 2003. [Cr80] Croke, C.: Some isoperimetric inequalities and eigenvalue estimates. Ann. Sci. Ec´ ole Norm. Sup. 13 (1980), 519–535. [Cr88] Croke, C.: An isoembolic pinching theorem. Invent. Math. 92 (1988), 385–387. [CK03] Croke, C.; Katz, M.: Universal volume bounds in Riemannian mani- folds, Surveys in Differential Geometry VIII, Lectures on Geometry and Topology held in honor of Calabi, Lawson, Siu, and Uhlenbeck at Har- vard University, May 3- 5, 2002, edited by S.T. Yau (Somerville, MA: International Press, 2003.) pp. 109 - 137. See arXiv:math.DG/0302248 [DaS98] Danilov, V. I.; Shokurov, V. V.: Algebraic curves, algebraic manifolds and schemes. Translated from the 1988 Russian original by D. Coray and V. N. Shokurov. Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences [Algebraic geometry. I, Encyclopaedia Math. Sci., 23, Springer, Berlin, 1994]. Springer-Verlag, Berlin, 1998. [Do79] Dombrowski, P.: 150 years after Gauss’ “Disquisitiones generales circa superficies curvas”. With the original text of Gauss. Ast´erisque, 62. Soci´et´e Math´ematique de France, Paris, 1979. [DR03] Dranishnikov, A.; Rudyak, Y.: Examples of non-formal closed (k 1)- connected manifolds of dimensions 4k 1, Proc. Amer. Math.−Soc. 133 (2005), no. 5, 1557-1561. See arXiv:math.AT/030≥ − 6299. [FK92] Farkas, H. M.; Kra, I.: Riemann surfaces. Second edition. Graduate Texts in Mathematics, 71. Springer-Verlag, New York, 1992. 94 BIBLIOGRAPHY

[Fe69] Federer, H.: Geometric Measure Theory. Grundlehren der mathematis- chen Wissenschaften, 153. Springer-Verlag, Berlin, 1969. [Fe74] Federer, H. Real flat chains, cochains, and variational problems. Indiana Univ. Math. J. 24 (1974), 351–407. [FHL98] Felix, Y.; Halperin, S.; Lemaire, J.-M.: The rational LS category of products and of Poincar´e duality complexes. Topology 37 (1998), no. 4, 749–756. [Fer77] Ferrand, J.: Sur la regularite des applications conformes, C.R. Acad.Sci. Paris Ser.A-B 284 (1977), no.1, A77-A79. [Fr99] Freedman, M.: Z2-Systolic-Freedom. Geometry and Topology Mono- graphs 2, Proceedings of the Kirbyfest (J. Hass, M. Scharlemann eds.), 113–123, Geometry & Topology, Coventry, 1999. [Fu16] Funk, P.: Uber eine geometrische Anwendung der Abelschen Integral- gleichung. Math. Ann. 77 (1916), 129-135. [GaHL04] Gallot, S.; Hulin, D.; Lafontaine, J.: Riemannian geometry. Third edi- tion. Universitext. Springer-Verlag, Berlin, 2004. [Ga71] Ganea, T.: Some problems on numerical homotopy invariants. Sympo- sium on Algebraic Topology (Battelle Seattle Res. Center, Seattle Wash., 1971), pp. 23–30. Lecture Notes in Math., 249, Springer, Berlin, 1971. [GG92] G´omez-Larranaga,˜ J.; Gonz´alez-Acuna,˜ F.: Lusternik-Schnirel’mann category of 3-manifolds. Topology 31 (1992), no. 4, 791–800. [Gre67] Greenberg, M. Lectures on Algebraic Topology. W.A. Benjamin, Inc., New York, Amsterdam, 1967. [GHV76] Greub, W.; Halperin, S.; Vanstone, R.: Connections, curvature, and cohomology. Vol. III: Cohomology of principal bundles and homogeneous spaces, Pure and Applied Math., 47, Academic Press, New York-London, 1976. [GriH78] Griffiths, P.; Harris, J.: Principles of algebraic geometry. Pure and Ap- plied Mathematics. Wiley-Interscience [John Wiley & Sons], New York, 1978. [Gro81a] Gromov, M.: Structures m´etriques pour les vari´et´es riemanniennes. (French) [Metric structures for Riemann manifolds] Edited by J. La- fontaine and P. Pansu. Textes Math´ematiques, 1. CEDIC, Paris, 1981. [Gro81b] Gromov, M.: Volume and bounded cohomology. Publ. IHES. 56 (1981), 213–307. [Gro83] Gromov, M.: Filling Riemannian manifolds, J. Diff. Geom. 18 (1983), 1-147. [Gro96] Gromov, M.: Systoles and intersystolic inequalities, Actes de la Table Ronde de G´eom´etrie Diff´erentielle (Luminy, 1992), 291–362, S´emin. Congr., 1, Soc. Math. France, Paris, 1996. www.emis.de/journals/SC/1996/1/ps/smf sem-cong 1 291-362.ps.gz [Gro99] Gromov, M.: Metric structures for Riemannian and non-Riemannian spaces, Progr. in Mathematics, 152, Birkh¨auser, Boston, 1999. [GroH81] Gross, B. H.; Harris, J.: Real algebraic curves. Ann. Sci. Ecole Norm. Sup. (4) 14 (1981), no. 2, 157–182. [GruL87] Gruber, P.; Lekkerkerker, C.: Geometry of numbers. Second edition. North-Holland Mathematical Library, 37. North-Holland Publishing Co., Amsterdam, 1987. BIBLIOGRAPHY 95

[Hat02] Hatcher, A.: Algebraic topology. Cambridge University Press, Cam- bridge, 2002. [He86] Hebda, J. The collars of a Riemannian manifold and stable isosystolic inequalities. Pacific J. Math. 121 (1986), 339-356. [Hel99] Helgason, S.: The Radon transform. 2nd edition. Progress in Mathemat- ics, 5. Birkh¨auser, Boston, Mass., 1999. [Hem76] Hempel, J.: 3-Manifolds. Ann. of Math. Studies 86, Princeton Univ. Press, Princeton, New Jersey 1976. [Hil03] Hillman, J. A.: Tits alternatives and low dimensional topology. J. Math. Soc. Japan 55 (2003), no. 2, 365–383. [Hi04] Hillman, J. A.: An indecomposable PD3-complex: II, Algebraic and Geometric Topology 4 (2004), 1103-1109. [Iv02] Ivanov, S.: On two-dimensional minimal fillings, St. Petersbg. Math. J. 13 (2002), no. 1, 17–25. [IK04] Ivanov, S.; Katz, M.: Generalized degree and optimal Loewner-type in- equalities, Israel J. Math. 141 (2004), 221-233. arXiv:math.DG/0405019 [Iw02] Iwase, N.: A∞-method in Lusternik-Schnirelmann category. Topology 41 (2002), no. 4, 695–723. [Jo48] John, F. Extremum problems with inequalities as subsidiary conditions. Studies and Essays Presented to R. Courant on his 60th Birthday, Jan- uary 8, 1948, 187–204. Interscience Publishers, Inc., New York, N. Y., 1948. [Kato83] Katok, A.: Entropy and closed geodesics, Ergo. Th. Dynam. Sys. 2 (1983), 339–365. [Kato92] Katok, S.: Fuchsian groups. Chicago Lectures in Mathematics. Univer- sity of Chicago Press, Chicago, IL, 1992. [KatH95] Katok, A.; Hasselblatt, B.: Introduction to the modern theory of dynam- ical systems. With a supplementary chapter by Katok and L. Mendoza. Encyclopedia of Mathematics and its Applications, 54. Cambridge Uni- versity Press, Cambridge, 1995. [Ka83] Katz, M.: The filling radius of two-point homogeneous spaces, J. Diff. Geom. 18 (1983), 505-511. [Ka88] Katz, M.: The first diameter of 3-manifolds of positive scalar curvature. Proc. Amer. Math. Soc. 104 (1988), no. 2, 591–595. [Ka02] Katz, M. Local calibration of mass and systolic geometry. Geom. Funct. Anal. (GAFA) 12, issue 3 (2002), 598-621. See arXiv:math.DG/0204182 [Ka03] Katz, M.: Four-manifold systoles and surjectivity of period map, Com- ment. Math. Helv. 78 (2003), 772-786. See arXiv:math.DG/0302306 [Ka05] Katz, M.: Systolic inequalities and Massey products in simply-connected manifolds, Geometriae Dedicata, to appear. [KL04] Katz, M.; Lescop, C.: Filling area conjecture, optimal systolic inequali- ties, and the fiber class in abelian covers. Proceedings of conference and workshop in memory of R. Brooks, held at the Technion, Israel Math- ematical Conference Proceedings (IMCP), Contemporary Math., Amer. Math. Soc., Providence, R.I. (to appear). See arXiv:math.DG/0412011 [KR04] Katz, M.; Rudyak, Y.: Lusternik-Schnirelmann category and systolic category of low dimensional manifolds. Communications on Pure and Applied Mathematics, to appear. See arXiv:math.DG/0410456 96 BIBLIOGRAPHY

[KR05] Katz, M.; Rudyak, Y.: Bounding volume by systoles of 3-manifolds. See arXiv:math.DG/0504008 [KatzRS06] Katz, M.; Rudyak, Y.; Sabourau, S.: Systoles of 2-complexes, Reeb graph, and Grushko decomposition. International Math. Research No- tices 2006 (2006). Art. ID 54936, pp. 1–30. See arXiv:math.DG/0602009 [KS04] Katz, M.; Sabourau, S.: Hyperelliptic surfaces are Loewner, Proc. Amer. Math. Soc., to appear. See arXiv:math.DG/0407009 [KS05] Katz, M.; Sabourau, S.: Entropy of systolically extremal surfaces and asymptotic bounds, Ergodic Theory and Dynamical Systems, 25 (2005). See arXiv:math.DG/0410312 [KS06] Katz, M.; Sabourau, S.: An optimal systolic inequality for CAT(0) met- rics in genus two, preprint. See arXiv:math.DG/0501017 [KSV05] Katz, M.; Schaps, M.; Vishne, U.: Logarithmic growth of sys- tole of arithmetic Riemann surfaces along congruence subgroups. See arXiv:math.DG/0505007 [Kl82] Klein, F.: Ub¨ er Riemanns Theorie der algebraischen Funktionen und ihrer Integrale. – Eine Erg¨anzung der gew¨ohnlichen Darstellungen. Leipzig: B.G. Teubner 1882. [Ko87] Kodani, S.: On two-dimensional isosystolic inequalities, Kodai Math. J. 10 (1987), no. 3, 314–327. [Kon03] Kong, J.: Seshadri constants on Jacobian of curves. Trans. Amer. Math. Soc. 355 (2003), no. 8, 3175–3180. [Ku78] Kung, H. T.; Leiserson, C. E.: Systolic arrays (for VLSI). Sparse Matrix Proceedings 1978 (Sympos. Sparse Matrix Comput., Knoxville, Tenn., 1978), pp. 256–282, SIAM, Philadelphia, Pa., 1979. [LaLS90] Lagarias, J.C.; Lenstra, H.W., Jr.; Schnorr, C.P.: Bounds for Korkin- Zolotarev reduced bases and successive minima of a lattice and its recip- rocal lattice. Combinatorica 10 (1990), 343–358. [La75] Lawson, H.B. The stable homology of a flat torus. Math. Scand. 36 (1975), 49–73. [Leib] Leibniz, G. W.: La naissance du calcul diff´erentiel. (French) [The birth of differential calculus] 26 articles des Acta Eruditorum. [26 articles of the Acta Eruditorum] Translated from the Latin and with an introduc- tion and notes by Marc Parmentier. With a preface by Michel Serres. Mathesis. Librairie Philosophique J. Vrin, Paris, 1989. [Li69] Lichnerowicz, A.: Applications harmoniques dans un tore. C.R. Acad. Sci., S´er. A 269, 912–916 (1969). [Lo71] Loewner, C.: Theory of continuous groups. Notes by H. Flanders and M. Protter. Mathematicians of Our Time 1, The MIT Press, Cambridge, Mass.-London, 1971. [Ma69] Macbeath, A. M.: Generators of the linear fractional groups. 1969 Num- ber Theory (Proc. Sympos. Pure Math., Vol. XII, Houston, Tex., 1967) pp. 14–32 Amer. Math. Soc., Providence, R.I. [Mi95] Miranda, R.: Algebraic curves and Riemann surfaces. Graduate Stud- ies in Mathematics, 5. American Mathematical Society, Providence, RI, 1995. [Ma79] Manning, A.: Topological entropy for geodesic flows, Ann. of Math. (2) 110 (1979), 567–573. BIBLIOGRAPHY 97

[Ma69] May, J.: Matric Massey products. J. Algebra 12 (1969) 533–568. [MS86] Milman, V. D.; Schechtman, G.: Asymptotic theory of finite-dimensional normed spaces. With an appendix by M. Gromov. Lecture Notes in Math- ematics, 1200. Springer-Verlag, Berlin, 1986. [MilH73] Milnor, J.; Husemoller, D.: Symmetric bilinear forms. Springer, 1973. [MiTW73] Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald: Gravi- tation. W. H. Freeman and Co., San Francisco, Calif., 1973. [Mo86] Moser, J. Minimal solutions of variational problems on a torus. Ann. Inst. H. Poincar´e-Analyse non lin´eaire 3 (1986), 229–272. [MS92] Moser, J.; Struwe, M. On a Liouville-type theorem for linear and non- linear elliptic differential equations on a torus. Bol. Soc. Brasil. Mat. 23 (1992), 1–20. [NR02] Nabutovsky, A.; Rotman, R.: The length of the shortest closed geodesic on a 2-dimansional sphere, Int. Math. Res. Not. 2002:23 (2002), 1211- 1222. [OR01] Oprea, J.; Rudyak, Y.: Detecting elements and Lusternik-Schnirelmann category of 3-manifolds. Lusternik-Schnirelmann category and related topics (South Hadley, MA, 2001), 181–191, Contemp. Math., 316, Amer. Math. Soc., Providence, RI, 2002. arXiv:math.AT/0111181 [Pa04] Parlier, H.: Fixed-point free involutions on Riemann surfaces. See arXiv:math.DG/0504109 [PP03] Paternain, G. P.; Petean, J.: Minimal entropy and collapsing with cur- vature bounded from below. Invent. Math. 151 (2003), no. 2, 415–450. [Pu52] Pu, P.M.: Some inequalities in certain nonorientable Riemannian mani- folds, Pacific J. Math. 2 (1952), 55–71. [Ri1854] B. Riemann, Hypoth`eses qui servent de fondement a` la g´eom´etrie, in Oeuvres math´ematiques de Riemann. J. Gabay, 1990. [RT90] Rosenbloom, M.; Tsfasman, M.: Multiplicative lattices in global fields. Invent. Math. 101 (1990), no. 3, 687–696. [Ru87] Rudin, W.: Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987. [Ru99] Rudyak, Y.: On category weight and its applications. Topology 38, 1, (1999), 37–55. [RT00] Rudyak, Y.; Tralle, A.: On Thom spaces, Massey products, and non- formal symplectic manifolds. Internat. Math. Res. Notices 10 (2000), 495–513. [Sa04] Sabourau, S.: Systoles des surfaces plates singuli`eres de genre deux, Math. Zeitschrift 247 (2004), no. 4, 693–709. [Sa05] Sabourau, S.: Entropy and systoles on surfaces, preprint. [Sa06] Sabourau, S.: Systolic volume and minimal entropy of aspherical mani- folds, preprint. [Sak88] Sakai, T.: A proof of the isosystolic inequality for the Klein bottle. Proc. Amer. Math. Soc. 104 (1988), no. 2, 589–590. [SS03] Schmidt, Thomas A.; Smith, Katherine M.: Galois orbits of principal congruence Hecke curves. J. London Math. Soc. (2) 67 (2003), no. 3, 673–685. [Sen91a] Senn, W. Strikte Konvexit¨at fur¨ Variationsprobleme auf dem n- dimensionalen Torus. manuscripta math. 71 (1991), 45–65. 98 BIBLIOGRAPHY

[Sen91b] Senn, W. Ub¨ er Mosers regularisiertes Variationsproblem fur¨ mini- male Bl¨atterungen des n-dimensionalen Torus. J. Appl. Math. Physics (ZAMP) 42 (1991), 527–546. [Ser73] Serre, J.-P.: A course in arithmetic. Translated from the French. Gradu- ate Texts in Mathematics, No. 7. Springer-Verlag, New York-Heidelberg, 1973. [Sik05] Sikorav, J.: Bounds on primitives of differential forms and cofilling in- equalities. See arXiv:math.DG/0501089 [Sin78] Singhof, W.: Generalized higher order cohomology operations induced by the diagonal mapping. Math. Z. 162, 1, (1978), 7–26. [Sm62] Smale, S.: On the structure of 5-manifolds. Ann. of Math. (2) 75 (1962), 38–46. [Sp79] Spivak, M.: A comprehensive introduction to differential geometry. Vol. 4. Second edition. Publish or Perish, Inc., Wilmington, Del., 1979. [St00] Streit, M.: Field of definition and Galois orbits for the Macbeath- Hurwitz curves. Arch. Math. (Basel) 74 (2000), no. 5, 342–349. [Tr93] Tralle, A.: On compact homogeneous spaces with nonvanishing Massey products, in Differential geometry and its applications (Opava, 1992), pp. 47–50, Math. Publ. 1, Silesian Univ. Opava, 1993. [Tu47] Tutte, W.: A family of cubical graphs. Proc. Cambridge Philos. Soc. 43 (1947), 459–474. [Wa83] Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics, 94. Springer-Verlag, New York- Heidelberg-Berlin, 1983. [We55] Weatherburn, C. E.: Differential geometry of three dimensions. Vol. 1. Cambridge University Press, 1955. [50] White, B.: Regularity of area-minimizing hypersurfaces at bound- aries with multiplicity. Seminar on minimal submanifolds (E. Bombieri ed.), 293-301. Annals of Mathematics Studies 103. Princeton University Press, Princeton N.J., 1983. [Wr74] Wright, Alden H.: Monotone mappings and degree one mappings be- tween P L manifolds. Geometric topology (Proc. Conf., Park City, Utah, 1974), pp. 441–459. Lecture Notes in Math., Vol. 438, Springer, Berlin, 1975. [Ya88] Yamaguchi, T.: Homotopy type finiteness theorems for certain precom- pact families of Riemannian manifolds, Proc. Amer. Math. Soc. 102 (1988), 660–666.