Riemannian geometry, peter petersen

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Riemannian Geometry With 60 Illustrations Springer Peter Petersen Department of Mathematics University of California, Los Angeles Los Angeles, CA 90095-1555 USA Editorial Board S Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA F.W Gehring Mathematics Department East Hall University of Michigan Ann Arbor, Ml48109 USA K.A Ribet Department of Mathematics University of California at Berkeley Berkeley, CA 94720-3840 USA Mathematics Subject Classification (1991): 53-01, 53C20 Library of Congress Cataloging-in-Publication Data Petersen, Peter, 1962Riemannian geometry /Peter Petersen p em -(Graduate texts in mathematics; 171) Includes bibliographical references (p - ) and index ISBN 978-1-4757-6434-5 (eBook) ISBN 978-1-4757-6436-9 DOI 10.1007/978-1-4757-6434-5 I Geometry, Riemannian QA649.P386 1997 516.3'73-dc21 I Title II Series 97-5786 Printed on acid-free paper © 1998 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc in 1998 Softcover reprint of the hardcover 1st edition 1998 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Production managed by Francine McNeill; manufacturing supervised by Johanna Tschebull Photocomposed copy prepared from the author's LaTeX files using Springer's svsing.sty macro 98765432 I ISBN 978-1-4757-6436-9 SPIN 10572067 To my wife, Laura Preface This book is meant to be an introduction to Riemannian geometry The reader is assumed to have some knowledge of standard manifold theory, including basic theory of tensors, forms, and Lie groups At times we shall also assume familiarity with algebraic topology and de Rham cohomology Specifically, we recommend that the reader is familiar with texts like [14] or[76, vol 1] For the readers who have only learned something like the first two chapters of [65], we have an appendix which covers Stokes' theorem, Cech cohomology, and de Rham cohomology The reader should also have a nodding acquaintance with ordinary differential equations For this, a text like [59] is more than sufficient Most of the material usually taught in basic Riemannian geometry, as well as several more advanced topics, is presented in this text Many of the theorems from Chapters to 11 appear for the first time in textbook form This is particularly surprising as we have included essentially only the material students ofRiemannian geometry must know The approach we have taken deviates in some ways from the standard path First and foremost, we not discuss variational calculus, which is usually the sine qua non of the subject Instead, we have taken a more elementary approach that simply uses standard calculus together with some techniques from differential equations We emphasize throughout the text the importance of using the correct type of coordinates depending on the theoretical situation at hand First, we develop our substitute for the second variation formula by using adapted coordinates These are coordinates naturally associated to a distance function If, for example, we use the function that measures the distance to a point, then the adapted coordinates are nothing but polar coordinates Next, we have exponential coordinates, which are of fundamental importance in showing that distance functions are smooth Then dis- vm Preface tance coordinates are used first to show that distance-preserving maps are smooth, and then later to give good coordinate systems in which the metric is sufficiently controlled so that one can prove, say, Cheeger's finiteness theorem Finally, we have harmonic coordinates These coordinates have some magic properties One in particular is that in such coordinates the Ricci curvature is essentially the Laplacian of the metric Our motivation for this treatment has been that examples become a natural and integral part of the text rather than a separate item that much too often is forgotten Another desirable by-product has been that one actually gets the feeling that gradients, Hessians, Laplacians, curvatures, and many other things are actually computable Often these concepts are simply abstract notions that are pushed around for fun From a more physical viewpoint, the reader will get the idea that we are simply using the Hamilton-Jacobi equations rather than the Euler-Lagrange equations to develop Riemannian geometry (see [4] for an explanation of these matters) It is simply a matter of taste which path one wishes to follow, but surprisingly, the Hamilton-Jacobi approach has never been tried systematically in Riemannian geometry The book can be divided into five imaginary parts: Part 1: Tensor geometry, consisting of Chapters to Part II: Classical geodesic geometry, consisting of Chapters and Part III: Geometry ala Bochner and Cartan, consisting of Chapters and Part IV: Comparison geometry, consisting of Chapters to 11 Appendices: de Rham cohomology, principal bundles, and spinors Chapters to give a pretty complete picture of some of the most classical results in Riemannian geometry, while Chapters to 11 explain some of the more recent developments in Riemannian geometry The individual chapters contain the following material: Chapter 1: Riemannian manifolds, isometries, immersions, and submersions are defined Homogeneous spaces and covering maps are also briefly mentioned We have a discussion on various types of warped products, leading to an elementary account of why the Hopffibration is also a Riemannian submersion Chapter 2: Many of the tensor constructions one needs on Riemannian manifolds are developed First the Riemannian connection is defined, and it is shown how one can use the connection to define the classical notions of Hessian, Laplacian, and divergence on Riemannian manifolds We proceed to define all of the important curvature concepts and discuss a few simple properties Aside from these important tensor concepts, we also develop several important formulas that relate curvature and the underlying metric These formulas are to some extent our replacement for the second variation formula The chapter ends with a short section where such Preface 1x tensor operations as contractions, type changes, and inner products are briefly discussed Chapter 3: First, we set up some general situations where it is possible to compute the curvature tensor The rest of the chapter is then devoted to carrying out this program in several concrete situations The curvature tensor of spheres, product spheres, warped products, and doubly warped products is computed This is used to exhibit some interesting examples that are Ricci fiat and scalar fiat In particular, we explain how the Riemannian analogue of the Schwarzschild metric can be constructed Several different models of hyperbolic spaces are mentioned Finally, we compute the curvatures of the Berger spheres and use this information as our basis for finding the curvatures of the complex projective plane Chapter 4: Here we concentrate on the special case where the Riemannian manifold is a hypersurface in Euclidean space In this situation, one gets some special relations between the curvatures We give examples of simple Riemannian manifolds that cannot be represented as hypersurface metrics Finally, we give a brief introduction to the Gauss-Bonnet theorem and its generalization to higher dimensions Chapter 5: The remaining foundational topics for Riemannian manifolds are developed in this chapter These include parallel translation, geodesics, Riemannian manifolds as metric spaces, exponential maps, geodesic completeness versus metric completeness, and maximal domains on which the exponential map is an embedding Chapter 6: Some of the classical results we prove here are: classification of simply connected space forms, the Hadamard-Cartan theorem, Preissmann's theorem, Cartan's center of mass construction in nonpositive curvature and why it shows that the fundamental group of such spaces is torsion free, Bonnet's diameter estimate, and Synge's theorem Chapter 7: Many of the classical and more recent results that arise from the Bochner technique are explained We look at Killing fields and harmonic 1-forms as Bochner did, and finally, discuss some generalizations to harmonic p-forms For the more advanced audience, we have developed the language of Clifford multiplication for the study of p-forms, as we feel that it is an important way of treating this material The last section contains some more exotic but also profound situations where the Bochner technique is applied to the curvature tensor These last two sections can easily be skipped in a more elementary course The Bochner technique gives many nice bounds on the topology of closed manifolds with nonnegative curvature In the spirit of comparison geometry, we show how Betti numbers of nonnegatively curved spaces are bounded by the prototypical compact fiat manifold: the torus x Preface The importance of the Bochner technique in Riemannian geometry cannot be sufficiently emphasized It seems that time and again, when people least expect it, new important developments come out of this simple philosophy Chapter 8: Part ofthe theory of symmetric spaces and holonomy is developed The standard representations of symmetric spaces as homogeneous spaces and via Lie algebras are explained We prove Cartan's existence theorem for isometries We explain how one can compute curvatures in general and make some concrete calculations on several of the Grassmann manifolds including complex projective space Having done this, we define holonomy for general manifolds, and discuss the de Rham decomposition theorem and several corollaries of it The above examples are used to give an idea of how one can classifY symmetric spaces Also, we show in the same spirit why symmetric spaces of (non)compact type have (nonpositive) nonnegative curvature operator Finally, we present a brief overview of how holonomy and symmetric spaces are related with the classification of holonomy groups This is used in a grand synthesis, with all that has been learned up to this point, to give Gallot and Meyer's classification of compact manifolds with nonnegative curvature operator A few things from Chapter are used in Chapter 8, namely Myers' theorem and the splitting theorem However, their use is inessential, and they are there to tie this material together with some of the more geometrical constructions that come later Chapter 9: Manifolds with lower Ricci curvature bounds are investigated in further detail First, we discuss volume comparison and its uses for Cheng's maximal diameter theorem Then we investigate some interesting relationships between Ricci curvature and fundamental groups The strong maximum principle for continuous functions is developed This result is first used in a warm-up exercise to give a simple proof of Cheng's maximal diameter theorem We then proceed to prove the Cheeger-Gromoll splitting theorem and discuss its consequences for manifolds with nonnegative Ricci curvature Chapter 10: Convergence theory is the main focus of this chapter First, we introduce the weakest form of convergence: Gromov-Hausdorff convergence This concept is often useful in many contexts as a way of getting a weak form of convergence The real object is then to figure out what weak convergence implies, given some stronger side conditions There is a section which breezes through Holder spaces, Schauder's elliptic estimates, and harmonic coordinates To facilitate the treatment of the stronger convergence ideas, we have introduced a norm concept for Riemannian manifolds We hope that these norms will make the subject a little more digestible The main idea of this chapter is to prove the Cheeger-Gromov convergence theorem, which is called the Convergence Theorem of Riemannian Geometry, and Anderson's generalizations of this theorem to manifolds with bounded Ricci curvature 418 Appendix C Spinors it is flat outside a compact set One can then refine these results to situations where one has certain types of curvature decay and various other topological conditions at infinity The interested reader can consult Greene's article in [46] for more on this story The idea behind the positive mass conjecture is that gravity cannot be isolated, i.e., the gravitational effects of a massive body can be measured everywhere in the universe While we haven't developed the machinery for accurately defining and describing this here (see [62] for the most readable account of the physics behind this), we have presented some Riemannian analogues that are very similar in nature The idea that gravity always attracts is translated into seal 2: 0, and the isolation phenomenon is that the space becomes flat outside a compact set, thus making it impossible to tell what happens in the compact region The further generalizations we mentioned give even stronger results along the line that scalar curvature (gravity) cannot decay too fast, depending on the dimension There is one more result that is easy to state, but whose proof unfortunately is not nearly so accessible The theorem was established by Schoen-Yau in low dimensions using complicated analytical machinery and by Gromov-Lawson using spin geometry Theorem 4.6 Any metric on the n-torus Tn with nonnegative scalar curvature is flat From the Bochner technique we know that this is true for any metric with nonnegative Ricci curvature Unfortunately, a similar argument using harmonic spinors doesn't seem to work Even though there are plenty of harmonic forms, there doesn't seem to be a way of extracting harmonic spinors Note that we need only one nontrivial spinor in order to conclude that the metric is Ricci flat Instead, one must resort to completely different techniques The reader can find a treatment in [54], but even then it is necessary to consult other papers to get the complete proof This theorem actually holds for a large class of manifolds, including all closed manifolds that admit metrics with nonpositive sectional curvature C.5 Further Study We have already mentioned the books [54] and [84] as sources for more in-depth discussions of spinors The first is a very exhaustive guide, while the latter gives a very nice, quick overview of the irreducible Clifford representations and the Bochner technique for spinors Both of these books use a slightly different notation than ours Namely, they think of the Clifford algebra as being the alternating algebra of multivectors rather than the algebra of forms It is our feeling that the approach used here is preferable in many ways First and foremost, all our formulae are invariant under general frames rather than just orthonormal frames, thus opening up the way for representing the theory in coordinates For the reader who is interested C.5 Further Study 419 in learning about spinors and index theory, the text [79, vol II] might be the best place to start The above-mentioned references [9] and [55] for the positive mass conjecture are also good for further study The reader might also wish to look at the very readable account of spin representations in [83] The reader who wishes to get a feeling for how spinors are used in physics can start with the comprehensive text [62] For some of the exciting new developments in 4-manifold theory that use spin geometry, we refer to [64] This book gives a basic account of spin geometry and how it is used to construct the Seiberg-Witten invariants Some of the important developments in this subject are intimately related to the things we have discussed above References [1] C.B Allendoerfer and A Weil, The Gauss-Bonnet theorem for Riemannian polyhedra, TAMS 53 (1943) 101-129 [2] M Anderson, Short Geodesics and Gravitationallnstantons, J Diff Geo 31 (1990) 265-275 [3] M Anderson, Metrics of positive Ricci curvature with large diameter, Manu Math 68 (1990) 405-415 [4] V.I Amol'd, Mathematical methods of classical mechanics, New York: SpringerVerlag, 1989 [5] W Ballmann, Non-positively curved manifolds ofhigher rank, Ann Math 122 (1985) 597-609 [6] W BaUmann, Spaces of non-positive curvature, Basel: Birkhiiuser, 1995 [7] W BaUmann and P Eberlein, Fundamental groups of manifolds of non-positive curvature, J Diff Geo 25 (1987) 1-22 [8] W BaUmann, V Schroeder and M Gromov, Manifolds of non-positive curvature, Boston: Birkhiiuser, 1985 [9] R Bartnik, The mass ofan asymptotically fiat manifold, Comm Pure Appl Math 39 (1986), 661-693 [10] M Berger, Riemannian geometry during the second half of the twentieth century, to appear [ 11] A.L Besse, Einstein Manifolds, Berlin-Heidelberg: Springer-Verlag, 1978 422 References [12] A.L Besse, Manifolds all ofwhose geodesics are closed, Berlin-Heidelberg: SpringerVerlag, 1987 [13] R.L Bishop and R.J Crittenden, Geometry ofManifolds, New York: Academic Press, 1964 [14] R.L Bishop and S.l Goldberg, Tensor analysis on manifolds, Dover, 1980 [15] R Bott and L.W Tu, Differential forms in algebraic topology, New York: SpringerVerlag, 1982 [16] K Bums and R Spatzier, On topological Tits buildings and their classification, IHES Publ Math 65 (1987) 5-34 [17] J.C Cantrell, ed., Geometric Topology, New York-London: Academic Press, 1979 [18] M.P Carmo, Differential forms and applications, Berlin-Heidelberg: Springer Verlag, 1994 [19] M.P Carmo, Riemannian Geometry, Boston: Birkhauser, 1993 [20] E Cartan, The theory of spinors, New York: Dover, 1966 [21] I Chavel, Riemannian Geometry, A Modern Introduction, New York: Cambridge University Press, 1995 [22] J Cheeger, Comparison and finiteness theorems for Riemannian manifolds, Ph D thesis, Princeton University [23] J Cheeger, Finiteness theorems for Riemannian manifolds, Am J Math 92 (1970), 61-75 [24] J Cheeger, Pinching theorems for a certain class of Riemannian manifolds, Am J Math 92 (1970), 807-834 [25] J Cheeger et al., Geometric Topology: Recent developments, LNM 1504, BerlinHeidelberg: Springer-Verlag, 1991 [26] J Cheeger and D Gromoll, The splitting theorem for manifolds ofnon-negative Ricci curvature, J Diff Geo (1971) 119-128 [27] S.S Chern, ed., Global Geometry and Analysis, 2nd edition, MAA Studies 27, Washington: Mathematical Association of America, 1989 [28] R Courant and D Hilbert, Methods ofMathematical Physics, vol II, New York: Wiley lnterscience, 1962 [29] B Chow and D Yang, Rigidity ofnon-negatively curved compact quaternionic-Kiihler manifolds, J Diff Geo 29 (1989) 361-372 [30] D DeTurk and J Kazdan, Some regularity theorems in Riemannian geometry, Ann scient Ec Norm Sup 14 (1981) 249-260 [31] R Edwards and R Kirby, Deformations of spaces of embeddings, Ann of Math 93 (1971), 63-88 References 423 [32] J.-H Eschenburg, Local convexity and non-negative curvature- Gromov's proofofthe sphere theorem, Invt Math 84 (1986) 507-522 [33] J.-H Eschenburg and E Heintze, An elementary proof of the Cheeger-Gromoll splitting theorem, Ann Glob Ana and Geo (1984) 141-151 [34] T Farrell and L Jones, Negatively curved manifolds with exotic smooth structures, J AMS (1989) 899-908 [35] K Fukaya, Hausdorff convergence of Riemannian manifolds and its applications, Advanced Studies in Pure Math 18-1 ( 1990) Recent topics in Differential and Analytic Geometry pp143-238 [36] S Gallot, Isoperimetric inequalities based on integral norms of Ricci curvature, Asterisque, 157-158, (1988) pp191-216 [37] S Gallot, D Hulin and J Lafontaine, Riemannian Geometry, Berlin-Heidelberg: Springer-Verlag, 1987 [38] S Gallot and D Meyer, Operateur de courbure et laplacien des formes differentielles d'une variete riemannianne, J Math Pores Appl 54 (1975), 259-284 [39] G Gibbons and S Hawking, Gravitational multi-instantons, Phys Lett B 78 (1978) 430-432 [40] D Gilbarg and N S Trudinger, Elliptic Partial Differential Equations ofSecond Order, 2nd edition, Berlin-Heidelberg: Springer-Verlag, 1983 [41] R Greene and S.T Yau, eds., Proc Symp Pure Math 54 vol3 (1994) [42] M Gromov, Manifolds of negative curvature, J Diff Geo 12 (1978) 223-230 [43] M Gromov, J Lafontaine and P Pansu, structures metriques pour les varietes riemannienness, Paris: Cedic/Femand Nathan, 1981 [44] M Gromov and W Thurston, Pinching constants for hyperbolic manifolds, Invt Math 89 (1987) 1-12 [45] K Grove, H Karcher, and E Rub, Group actions and curvature, Invt Math 23 ( 197 4}, 31-48 [46] K Grove and P Petersen, eds., Comparison Geometry, MSRI publications vol 30, New York: Cambridge University Press, 1997 [47] K Grove, P Petersen, and J.-Y Wu, Geometric finiteness theorems via controlled topology, Invt Math 99 (1990) 205-213, Erratum Invt Math 104 (1991) 221-222 [48] R.S Hamilton, The formation of singularities in the Ricci flow, Surveys in Diff Geo vol 2, International Press (1995) 7-136 [49] S Helgason, Differential Geometry, Lie Groups and Symmetric spaces, New YorkLondon: Academic Press, 1962 [50] J Jost, Riemannian Geometry and Geometric Analysis, Berlin-Heidelberg: SpringerVerlag, 1995 424 References [51] J Jost and H Karcher, Geometrische Methoden zur Gewinnung von a-pnonSchrankenfiir harmonische Abbildungen, Manu Math 19 (1982) 27-77 [52] S Kobayashi, Transformation Groups in Differential Geometry, Berlin-Heidelberg: Springer-Verlag, 1972 [53] S Kobayashi and K Nomizu, Foundations of Differential Geometry, vols I, II, New York: Wiley-Interscience, 1963 [54] H.B Lawson Jr and M.-L Michelsohn, Spin Geometry, Princeton: Princeton University Press, 1989 [55] J.M Lee and T.H Parker, The Yamabe problem, Bull AMS 17 (1987), 37-91 [56] A Lichnerowicz, Geometrie des groupes de transformations, Paris: Dunod, 1958 [57] A Lichnerowicz, Propagateurs et Commutateurs en relativite generate, Publ Math IHES 10 (1961) 293-343 [58] M.J Micallef and J.D Moore, Minimall-spheres and the topology of manifolds with positive curvature on totally isotropic 2-planes, Ann of Math 127 (1988), 199-227 [59] R.K Miller and A.N Michel, Ordinary differential equations, New York-London: Academic Press, 1982 [60] J.W Milnor, Morse Theory, Princeton: Princeton University Press, 1963 [61] J W Milnor and J.D Stasheff, Characteristic Classes, Princeton: Princeton University Press, 1974 [62] C.W Misner, K.S Thome and J.A Wheeler, Gravitation, New York: Freeman, 1973 [63] N Mok, The uniformization theorem for compact Kahler manifolds of non-negative holomorphic hi-sectional curvature, J Diff Geo 27 (1988), 179-214 [64] J W Morgan, The Seiberg-Witten equations and applications to the topology ofsmooth four-manifolds, Princeton: Princeton Univ Press, 1996 [65] B O'Neill, Semi-Riemannian Geometry, New York-London: Academic Press, 1983 [66] Y Otsu, On manifolds ofpositive Ricci curvature with large diameters, Math Z 206 (1991) 255-264 [67] M Ozaydm and G Walschap, Vector bundles with no soul, PAMS 120 (1994) 565-567 [68] G Perel'man, Alexandrov's spaces with curvatures bounded from below ll, preprint [69] S Peters, Cheeger's finiteness theorem for diffeomorphism classes of manifolds, J Reine Angew Math 349 (1984) 77-82 [70] P Petersen, S Shteingold and G Wei, Comparison geometry with integral curvature bounds, to appear in GAFA [71] P Petersen and G Wei, Relative volume comparison with integral Ricci curvature bounds, to appear in GAFA [72] G de Rham, Differentiable Manifolds, Berlin-Heidelberg: Springer-Verlag, 1984 425 [73] X Rong, The almost cyclicity of the fundamental groups of positively curved manifolds, Invt Math 126 (1996) 47-64 And Positive curvature, local and global symmetry, and fundamental groups, preprint, Rutgers, New Brunswick [74] X Rong, A Bochner Theorem and Applications, preprint To appear in Duke Math J [75] Y Shen and S.-h Zhu, Ricci curvature, minimal surfaces, and sphere theorems To appear in J Geo Anal [76] M Spivak, A Comprehensive Introduction to Differential Geometry, vols 1-V, Wilmington: Publish or Perish, 1979 [77] J J Stoker, Differential Geometry, New York: Wiley-Interscience, 1989 [78] S Tachibana, A theorem on Riemannian manifolds of positive curvature operator, Proc Japan Acad 50 (1974), 301-302 [79] M.E Taylor, Partial differential equations, vols 1-Ill, New York: Springer Verlag, 1996 [80] T Y Thomas, Riemann spaces of class one and their characterization, Acta Math 67 (1936) 169-211 [81] F Tricerri andL Vanhecke, C R Acad Sc Paris t.302 Series I no (1986) 233-235 [82] F.W Warner, Foundations of Differentiable Manifolds and Lie Groups, New York: Springer-Verlag, 1983 [83] H Weyl, The classical groups, Princeton: Princeton University Press, 1966 [84] H.H Wu, The Bochner Technique in Differential Geometry, Mathematical Reports vol 3, part 2, London: Harwood Academic Publishers, 1988 Index Angle comparison in negative curvature, 151 Arzela-Ascoli lemma, 279 Axis for an isometry, 152 Berger spheres, 6, 94, 101, 160, 277 computation of curvatures, 81 Betti number estimate by Bochner, 178 by Gallot-Meyer, 182 by Gromov, 342 by Gromov-Gallot, 253 Bianchi identities for the curvature form, 382 Bianchi's first identity, 26 Bianchi's second identity, 26 Bochner formula, 314 for forms, 188 for the curvature tensor, 191 Bochner technique for 1-forms, 173 for Killing fields, 166 for p-forms, 181 for spinors, 404 in general, 178 Bonnet's diameter estimate, 154 Bundles of frames, 136 over 2-sphere, 18, 85, 342, 359 Busemann function, 264 Cartan formalism, 56, 377 Cartan's theorem, 150 Center of mass, 150 Cheeger's lemma, 300 Cheng's maximal diameter theorem, 248,263 Clifford algebra, 401 Clifford bundle, 401 Clifford multiplication, 184 Codazzi equation, see Normal curvature equation Cohomology Cech, 366 de Rham, 170, 367 compactly supported, 372 Hodge, 173 Compact embedding, 283 Comparison estimates for Ricci curvature, 244 for sectional curvature, 140 Comparison geometry, 46 Completeness, 125 geodesic, 109 428 Index Completeness (cont.) metric, 114 of closed manifolds, 116 of Gromov-Hausdorff topology, 277 Conjugate point, 45, 127, 148 Conjugate radius estimate, 148 Connection affine, 22, 54 along curves, 104 form, 56, 98, 378 in local coordinates, 56 of hi-invariant metric, see metric on Euclidean space, 21 on Lie group, 54 on vector bundle, 178 representation in a frame, 56 Riemannian, 22 Constant curvature, 29 global classification, 137 local characterization, 123 Contractions, 50 Convergence of maps, Gromov-Hausdorff, 279 of pointed spaces, Gromov-Hausdorff, 279 of spaces Gromov-Hausdorff, 274 Hausdorff, 27 in Holder topology, 290 Convergence theorem of Anderson, 308 of Cheeger-Gromov, 300 Convexity radius, 148 Coordinates adapted, 42 Cartesian in Euclidean space, 21 on Riemannian manifold, 121 distance, 132, 299 exponential, 119 harmonic, 285 normal at a point, 54 polar in the plane, on Riemannian manifold, 121 , 123 representation of metric in, Covariant derivative, 23 in parallel frame, 106 of a tensor, 391 Covering space, see Riemannian Critical point estimate, 334 Critical point theory, 318 Curvature constant, see Constant curvature directional, 29, 37, 48 form, 56, 98, 380 Gauss, 96 in dimension 2, 31 in dimension 3, 31, 59 in local coordinates, 56 Isotropic, 60, 312 of a tensor, 28 of hi-invariant metric, 57 of product metric, see Product oftensors, 194,392 operator, 29, 59, 182, 190, 196, 343 classification of 2: 0, 237 on symmetric spaces, 232 representation in a frame, 56 Ricci, 30, 243, 305 in harmonic coordinates, 286 Riemannian, 26 scalar, 32, 183 sectional, 29, 48, 59, 137, 195, 317 Cut locus, 130 Degree of a map, 374 Dirac operator, 183 on forms, 186 on spinors, 404 Directional derivative, 20 Dirichlet problem, 285 Displacement function, 152 Distance function, 34, 41, 116, 120, 123 Divergence, 24, 54 Doubly warped products, see Products Eguchi-Hanson metric, 85 Einstein constant, 31 metric, 31, 228 notation, Elliptic estimates, 284 Elliptic operators, 284 Euclidean space, curvature of, 28 Euler characteristic, 97 Exponential map, 117 Index Lie group, 18 Exponential map comparison, 145 Extrinsic geometry, 36, 89 Fibration, 136 Finiteness theorem for diffeomorphism types, 297, 302 in positive curvature, 300 for fundamental groups, 255 for homotopy types, 351 Focal point, 45 Frame left invariant, 10 normal at a point, 23, 54 representation of metric in, 10 Frame bundle, 166,384 Framing, see Frame Frankel's theorem, 162 Functional distance, see Metric Fundamental equations for curvature, 34 of Riemannian geometry, 40 Fundamental theorem of convergence theory, 293 of hypersurface theory, 95 of Riemannian geometry, 21, 22 Gauss equation, see Tangential curvature equation Gauss lemma, 120 Gauss map, 90 Gauss-Bonnet theorem, 96 Geodesic, 106, 117, 120 Gradient, 20 Grassmannian compact as a symmetric space, 217 computation of curvatures, 219 hyperbolic as a symmetric space, 220 computation of curvatures, 222 Holder norms, 281 Hadamard theorem, 91 Hadamard-Cartan theorem, 46, 147 Harmonic function, 259 Hessian, 24 Hessian comparison, 143, 326 Hinge, 322 429 Hodge star operator, 171 Hodge theorem, 173 Holonomy, 226,414 of symmetric spaces, 235 Holonomy classification, 235 Homogeneousspace,5 completeness of, 134 k-point, 161 Hopf fibration, 4, 6, 15 Hopfproblem, 101, 169, 182,343 Hopf theorem, 97 Hopf-Rinow theorem, 125 Hyperbolic space, 77 as left-invariant metric, 80 as rotationally symmetric surface, 12 geodesics in, 111 isometry group of, 80 Minkowski space model, 78 Riemann's model, 77 Rotationally symmetric model, 77 upper half plane model, 77 Hypersurface in Euclidean space, 34, 89 in Riemannian manifold, 155 Index notation, 52 Injectivity radius, 130 lnjectivity radius estimate by Cheeger, 300 generalization of Cheeger's lemma, 315 in general, 149 in positive curvature, 159 Intrinsic geometry, 36, 89, 96 Invariant Cartan formalism, 389 Isometric immersion, see Riemannian Isometry distance-preserving, 132 Riemannian, see Riemannian Isometry group, 5, 165 of Euclidean space, of Hyperbolic space, 80 of the sphere, Isotropy group, Jacobi equation, 263 Jacobi field, 262 Killing field, 55, 164, 216 430 Index Killing frame, 200 Klein bottle, Koszul formula, 22 Kuratowski embedding, 278 Laplacian connection, 179 coordiante representation, 55 in harmonic coordinates, 286 on forms, 172 on functions, 24 on spinors, 404 Law of cosines, 324 Left invariant frame, 10 metric, see Metric Length comparison, 157, 161, 162 Length of curve in metric space, 135 in Riemannian manifold, 113 Lichnerowicz formula, 183, 408 Lie group, hi-invariant metric, see Metric geodesics of hi-invariant metric, 136 geodesics on, 112 Line,260 Lipschitz norm, see norm Local models, 314 Matrix inequalities, 47, 140 Maximum principle, 55,257 Metric ball, 113 hi-invariant, 18, 57, 112, 136, 271 as a symmetric space, 208, 225 distance, 112 Einstein, 31 functional, 116, 136 homogeneous, see Homogeneous space Kahler, 61, 203, 239, 240 left-invariant, local representation of, on frame bundle, 136 on tangent bundle, 136 rotationally symmetric, 12 computation of curvatures, 70 scalar flat, 72 Meyer-Vietoris sequence for de Rham cohomology, 367 Mixed curvature equation, see normal curvature equation Musical isomorphisms, 184 Myers' diameter estimate, 47, 245 Norm cm,a for functions, 283 for manifolds, 289 harmonic for manifolds, 303 Lipschitz for manifolds, 298 of tensors, 51 weak for manifolds, 315 weighted for manifolds, 314 Norm estimate using distance functions, 299, 302 using exponential maps, 299 using harmonic coordinates, 306 Normal curvature equation, 38, 67 in Euclidean space, 91 O'Neill's formula, 58 Obstructions for constant sectional curvature, 140 for negative curvature operator, 101 for negative sectional curvature, 147 for nonnegative sectional curvature, 342,343 for positive curvature operator, 101 , 182 for positive Ricci curvature, 183 for positive scalar curvature, 183 for positive sectional curvature, 159 for Ricci flatness, 267 Poincare duality, 370 Parallel along curve, 105 on manifold, 24 vector field, 59 Parallel curvature, 210 Pinching theorem for Ricci curvature, 311 for sectional curvature, 311 Index Poincare lemma for de Rham cohomology, 368 Positive mass conjecture, 416 Precompactness theorem for lower Ricci curvature bounds, 281 for spaces with bounded norm, 294 in Gromov-Hausdorff topology, 280 Preissmann's Theorem, 151 Principal bundles, 392 Product Cartesian, 17, 58 doubly warped, 13 computation of curvatures, 73 warped, 69 Product spheres computations of curvatures, 69 Projective space complex, 6, 17, 92 as a symmetric space, 222 computation of curvatures, 81, 223 holonomy of, 240 quatemionic, 240 real, Quarter pinching, 60, 331 Radial curvature equation, 37, 66 Rank,210,236 rigidity in nonpositive curvature, 237 Ray,260 de Rham's decomposition theorem, 228 de Rham's theorem, 369 Riemannian covering, 6, 146 embedding, immersion, in Euclidean space, 91 isometry, uniqueness of, 137 manifold, submersion, 4, 58, 133, 136 Scaling, 60 Schur's lemma, 32 Schwarzschild metric, 75 Second covariant derivative, 25 Second fundamental form, see Shape operator Segment, 114, 120, 122 431 characterization, 127 Segment domain, 127 Shape operator, 36 for hypersurface in Euclidean space, 91 for hypersurface in Riemannian manifold, 155 Soul theorem, 334 Sphere, as doubly warped product, 15 as rotationally symmetric metric, 14 as surface of revolution, 11 computation of curvatures, 69 geodesics on, 110 isometry group of, Sphere theorem Grove-Shiohama, 332 Rauch-Berger-Klingenberg, 331 Spin manifolds, 183 Spin structures, 396 Spinor bundles, 399 Spinors, 395 local representaion of, 403 parallel, 414 square of, 41 in dimension 3, 413 tensor square of, 409 Splitting theorem, 47, 261 Structural equations First, 379 Second,380 S U (2), see Berger spheres Subharmonic function, 259 Submetry, 133 Superharmonic function, 259 Surface of revolution, 10, 92 rotationally symmetric, 11 Symmetric space, 208, 212 computation of curvatures, 215 existence of isometries, 211 Synge's lemma, 46, 158 Tangential curvature equation, 37, 66 in Euclidean space, 91 Tensor bundles, 386 Topology manifold, 115 metric, 115 432 Index Toponogov comparison theorem, 323 Torus, 7, 17 Totally geodesic, 162 Triangle, 322 Type change, 48 Volume comparison absolute, 246 for cones, 270 relative, 247, 270 Volume form, 54, 246 Weak second derivatives, 257 Weitzenbock formula, 181 forforms, 187 for spinors, 404 Weyl tensor, 86 Graduate Texts in Mathematics continued from page ii 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 WIDTEHEAD Elements of Homotopy Theory ICARGAPOLOV/MERLZJAKOV Fundamentals of the Theory of Groups BOLLOBAS Graph Theory EDWARDS Fourier Series Vol I 2nd ed WELLS Differential Analysis on Complex Manifolds 2nd ed WATERHOUSE Introduction to Affine Group Schemes SERRE Local Fields WEIDMANN Linear Operators in Hilbert Spaces LANG Cyclotomic Fields II MAsSEY Singular Homology Theory FARKAs/KRA Riemann Surfaces 2nd ed STILLWELL Classical Topology and Combinatorial Group Theory 2nd ed HUNGERFORD Algebra DAVENPORT Multiplicative Number Theory 2nd ed HOCHSCHILD Basic Theory of Algebraic Groups and Lie Algebras lrrAKA Algebraic Geometry HECKE Lectures on the Theory of Algebraic Numbers BVRRis/SANKAPPANAVAR A Course in Universal Algebra WALTERS An Introduction to Ergodic Theory ROBINSON A Course in the Theory of Groups 2nd ed FoRSTER Lectures on Riemann Surfaces Borr/Tu Differential Forms in Algebraic Topology WASHINGTON Introduction to Cyclotomic Fields 2nd ed IRELAND/RosEN A Classical Introduction to Modem Number Theory 2nd ed EDWARDS Fourier Series Vol II 2nd ed VAN LINT Introduction to Coding Theory 2nd ed BROWN Cohomology of Groups PIERcE Associative Algebras LANG Introduction to Algebraic and Abelian Functions 2nd ed BR0NDSTED An Introduction to Convex Polytopes BEARDON On the Geometry of Discrete Groups 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 DIESTEL Sequences and Series in Banach Spaces DUBROVIN/FOMENKO/NOVIKOV Modem Geometry-Methods and Applications Part I 2nd ed WARNER Foundations of Differentiable Manifolds and Lie Groups SHIRYAEV Probability 2nd ed CoNWAY A Course in Functional Analysis 2nd ed KoBLITZ Introduction to Elliptic Curves and Modular Forms 2nd ed BR6cKERIToM DIECK Representations of Compact Lie Groups GRoVFJBENSON Finite Reflection Groups 2nd ed BERG/CHRISTENSEN/REsSEL Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions EDWARDS Galois Theory V ARADARAJAN Lie Groups, Lie Algebras and Their Representations LANG Complex Analysis 3rd ed DUBROVIN/FOMENKO/NoVIKov Modem Geometry-Methods and Applications Part II LANG SL2(R) SILVERMAN The Arithmetic of Elliptic Curves OLVER Applications of Lie Groups to Differential Equations 2nd ed RANGE Holomorphic Functions and Integral Representations in Several Complex Variables LEHTO Univalent Functions and Teichmiiller Spaces LANG Algebraic Number Theory HUSEMOLLER Elliptic Curves LANG Elliptic Functions KARATZAs/SHREVE Brownian Motion and Stochastic Calculus 2nd ed KoBLITZ A Course in Number Theory and Cryptography 2nd ed BERGER!GosnAUX Differential Geometry: Manifolds, Curves, and Surfaces KELLEY/SRINIVASAN Measure and Integral Vol I SERRE Algebraic Groups and Class Fields PEDERSEN Analysis Now 119 RoTMAN An Introduction to Algebraic Topology 120 ZIEMER Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation 121 LANG Cyclotomic Fields I and II Combined 2nd ed 122 REMMERT Theory of Complex Functions Readings in Mathematics 123 EBBINGHAUS/HERMES et al Numbers Readings in Mathematics 124 DUBROVIN/FOMENKO/NOVII