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On Geodesics in Riemannian Geometry Lectures on Geodesics Riemannian Geometry By M. Berger Lectures on Geodesics Riemannian Geometry By M. Berger No part of this book may be reproduced in any form by print, microfilm or any other means with- out written permission from the Tata Institute of Fundamental Research, Colaba, Bombay 5 Tata Institute of Fundamental Research Bombay 1965 Introduction The main topic of these notes is geodesics. Our aim is twofold. The first is to give a fairly complete treatment of the foundations of rieman- nian geometry through the tangent bundle and the geodesic flow on it, following the path sketched in [2] and [19]. We construct the canonical spray of a riemannian manifold (M, g) as the vector field G on T(M) defined by the equation 1 i(G) dα = dE, · −2 where i(G) dα denotes the interior product by G of the exterior deriva- · tive dα of the canonical form α on (M, g) (see (III.6)) and E the energy (square of the norm) on T(M). Then the canonical connection is intro- duced as the unique symmetric connection whose associated spray is G. The second is to give global results for riemannian manifolds which are subject to geometric conditions of various types; these conditions involve essentially geodesics. These global results are contained in Chapters IV, VII and VIII. Chapter IV contains first the description of the geodesics in a symmetric compact space of rank one (called here an S.C.-manifold) and the de- scription of Zoll’s surface (a riemannian manifold, homeomorphic to the two-dimensional sphere, non isometric to it and all of whose geodesics through every point are closed). Then we sketch results of Samelson and Bott to the effect that a riemannian manifold all of whose geodesics are closed has a cohomology ring close to that of an S.C.-manifold. In Chapter VII are contained the Hopf-Rinow theorem, the existence of a closed geodesic in a non-zero free homotopy class of a compact rie- iii iv Introduction mannian manifold, and the isometry between two simply connected and complete riemannian manifolds of the same constant sectional curva- ture. In Chapter VIII one will find theorems of Myers and Synge, the Gauss-Bonnet formula and a result on complete riemannian manifolds of non-positive curvature. Then come the theorem of L.W. Green, which asserts that, on the two-dimensional real-projective space, a riemannian structure all of whose geodesics are closed has to be isometric to the standard one, and the theorem of E. Hopf: on the two-dimensional torus, a riemannian structure all of whose geodesics are without conju- gate points has to be a flat one. As a counterpoint we have quoted the work of Busemann which shows that the theorems of Green and Hopf pertain to the realm of riemannian geometry, for they no longer hold good in G-spaces (see (VIII.10)). However, the result on complete man- ifolds with non-positive curvature is still valid in G-spaces. We have included in Chapter VIII theorems of Loewner and Pu, which are “isoperimetric” inequalities on the two dimensional torus and the two-dimensional real projective space (equality implying isometry with the standard riemannian structures on these manifolds). These re- sults do not involve geodesics explicitly, but have been included for their great geometric interest. One should also note that the results of Green, Hopf, Loewner and Pu are two-dimensional and so lead to interesting problems in higher dimensions. The tools needed for these results are developed in various chapters: Jacobi fields, sectional curvature, the second variation formula play an important role; see also the formulas in (VIII.4) and (VIII.8). The reference for calculus is [36]; references for differential and rie- mannian geometry are [14]; [16], [17], [18], [19], [21], [33], [35], and lectures notes of I.M.Singer and a seminar held at Strasbourg University. We have used some of these references without detailed acknowledge- ment. I am greatly indebted to N. Bourbaki, P. Cartier and J.L. Koszul for communication and permission to use ideas and results of theirs on connections. For most valuable help I am glad to thank here P. Cartier, J.L. Koszul, N. Kuiper and D. Lehmann. Contents Introduction iii 0 Preliminaries 1 1 .............................. 1 2 Forms........................... 6 3 Integration......................... 9 4 .............................. 14 5 Curves........................... 21 6 Flows ........................... 24 1 Sprays 27 1 Definition.......................... 27 2 Geodesics ......................... 29 3 Expressions for the spray in local coordinates . 31 4 The exponential map . 33 2 Linear connections 37 1 Linear connection . 37 2 Connection in terms of the local coordinates . 41 3 Covariant derivation . 43 4 The derivation law . 48 5 Curvature ......................... 53 6 Convexity ......................... 60 7 Parallel transport . 65 8 Jacobifields........................ 69 v vi Contents 3 Riemannian manifolds 81 2 Examples ......................... 85 3 Symmetricpairs...................... 87 4 TheS.C.-manifolds ..... ..... ...... .... 94 5 Volumes..........................100 6 The canonical forms α and dα ..............104 7 Theunitbundle ......................109 8 Expressions in local coordinates . 110 4 Geodesics 113 1 The first variation . 113 2 The canonical spray on an r.m.(M, g)...........117 3 First consequences of the definition . 122 4 Geodesics in a symmetric pair . 124 5 Geodesics in S.C.-manifolds . 127 6 Results of Samelson and Bott . 130 7 Expressions for G in local coordinates . 139 8 Zoll’ssurface .......................142 5 Canonical connection 151 2 Riemannian structures on T(M) and U(M)........156 3 Dg = 0...........................162 4 Consequences of Dg = 0 .................165 5 Curvature .........................167 6 Jacobi fields in an r.m. 169 6 Sectional Curvature 179 2 Examples .........................181 3 Geometric interpretation . 184 4 A criterion for local isometry . 187 5 Jacobi fields in symmetric pairs . 195 6 Sectional curvature of S.C.-manifolds . 197 7 Volumes of S.C.manifolds . 199 8 Ricci and Scalar curvature . 206 Contents vii 7 The metric structure 213 2 The metric structure . 221 3 Niceballs .........................223 4 Hopf-Rinow theorem . 230 5 A covering criterion . 235 6 Closed geodesics . 239 7 Manifolds with constant sectional curvature . 246 8 Some formulas and applications 249 1 The second variation formula . 249 2 Second variation versus Jacobi fields . 254 3 The theorems of Synge and Myers . 261 4 Aformula.........................266 5 Index of a vector field . 272 6 Gauss-Bonnet formula . 274 7 E.Hops’stheorem ... ..... ...... ..... ..280 8 Anotherformula.... ..... ...... ..... ..282 9 L.W.Green’s theorem . 288 10 Concerning G-spaces ...................292 11 Conformal representation . 294 12 The theorems of Loewner and Pu . 299 Chapter 0 Preliminaries 1 1 In this chapter we formulate certain notions connected with the notion of a manifold in a form we need later on and fix some notations and conventions. By means of a basis we identify a d-dimensional real vector space with set Rd of all d-tuples of real numbers with the standard vector V space structure and denote the element with 1 in the ith place and zeros elsewhere by e . Then e form a basis of Rd, we call it the canonical i { i} basis of Rd, and its dual basis (u1,..., ud) the canonical coordinate sys- tem in Rd. (For R(= R1) we set u1 = t). With this identification any real vector space becomes a manifold and this structure is independent of the basis chosen. A manifold will always be a C∞-manifold which is Hausdorff, para compact and of constant dimension d. Generally we denote it by M, a typical chart of it by (U, r) and local coordinates with respect to (U, r) by xi = ui r. Let us note that because of para compactness partitions ◦ of unity for M exist. We denote the tangent space of M at a by Ta(M) and define the tangent bundle T(M) of M to be (0.1.1) x = T(M) { } a[M x [Ta(M) ∈ ∈ and call elements of T(M) vectors of M. Then we have the natural 1 2 Preliminaries projection map pM (or simply p) from T(M) onto M which takes every vector of M at a onto a. 2 We denote the set of all maps (C∞-maps, differentiable maps) of a manifold M into a manifold N byD(M, N) and when N = R we write F(M) for D(M, R). Every f in D(M, N) induces a map of F(N) into F(M) defined by g g f → ◦ and this in turn a map f T of T(M) into T(N) defined by the equation (0.1.2) ( f T (x))(g) = x(g f ), g F(N). ◦ ∈ Then we have the following commutative diagram f T T(M) / T(N) (0.1.3) pM pN f M / N 1.4 T Let us note that if f restricted to Tm(M) is one-one then we can choose a local coordinate system (x1,..., xe) for N at n = f (m) such that xi f { ◦ } (i = 1,..., d) form a local coordinate system at m for M. Also if f T restricted to Tm(M) is onto Tn(N) then we can choose a local coordinate system (x1,..., xd) for M at m and a local coordinate system (y1,..., ye) for N at n such that xi = yi f , i = 1,..., 0. ◦ 1.5 We call a manifold N a sub manifold of M if 3 1) N M, ⊂ 2) the topology on N is induced by that on M, 1. 3 3) to each point p of N there is a chart (Up, rp) in the atlas of M such that for some positive integer k r (N U ) = (x ) r (U ) x + = ..
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