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Differential Geometry and Topology Mikhail G. Katz∗

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Differential Geometry and Topology Mikhail G. Katz∗ 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 2 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 introducef ang 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) f g 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; f g 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.
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