Bi bl iography. c.r. Acad. Sci. Paris 247 (1958) pp R. ABRAHAM, Lectures of Smale on Differential Topology, Columbia University, 1962

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1 Bi bl iography [Ab 62] [AbM 78] [Am 56] [Am 60] [Am 61] [APS 60] [ABP 75] [AH 59] [APS 75] [Be 58] [BGM 71] [BGV 92] R. ABRAHAM, Lectures of Smale on Differential Topology, Columbia University, 1962 R. ABRAHAM and J. MARSDEN, Foundations of Mechanics, second edition, Benjamin-Cummings, 1978 W. AMBROSE, Parallel translation of Riemannian curvature, Ann. of Math. 64 (1956) pp W. AMBROSE, The Cartan structural equations in classical Riemannian geometry, J. Indian Math. Soc. 24 (1960) pp W. AMBROSE, The index theorem in Riemannian geometry, Ann. of Math. 73 (1961) pp W. AMBROSE, R.S. PALAIS, and I.M. SINGER, Sprays, Acad. Brasileira de Ciencias 32 (1960) pp M. ATiYAH, R. BOTT, and V.K. PATODl, On the heat equation and the index theorem, Invent. Math. 19 (1973) pp ; Errata, Invent. Math. 28 (1975) pp M. ATiYAH and F. HIRZEBRUCH, Riemann-Roch theorems for differentiable manifolds, Bull. Amer. Math. Soc. 65 (1959) pp M. ATiYAH, V.K. PATODl, and I. SINGER, Spectral asymmetry and Riemannian geometry: I. Math. Proc. Camb. Phi/os. Soc. 77 (1975) pp M. BERGER, Sur certaines varietes Riemanniennes a courbure positive, c.r. Acad. Sci. Paris 247 (1958) pp M. BERGER, P. GAUDUCHON, and E. MAZET, Le Spectre d'une Variete Riemannienne, Lecture Notes in Mathematics 195, Springer-Verlag, 1971 N. BERLlNE, E. GETZLER, and M. VERGNE, Heat Kernels and Dirac Operators, Grundlehren der Math. Wiss. 298, Springer-Verlag, 1992

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3 BIBLIOGRAPHY 357 [Go 62] [Gr 61] [Gray 73] [GrV 79] [GriH 76] [GrKM 75] [Gros 64] [Gu 91] [Ha 1898] [He 61] [He 78] [He 84] [Hi 59] [HoR 31] [Ir 70] [Ke 55] [Ko 87] [KoN 63] [KoN 69] [Kos 57] [Kr 72] [La 61] S.I. GOLDBERG, Curvature and Homology, Academic Press, 1962 W. GRAEUB, Liesche Grupen und affin zusammenhangende Mannigfaltigkeiten, Acta Math. (1961) pp A. GRAY, The volume of small geodesic balls, Mich. J. Math. 20 (1973) pp A. GRAY and L. VANHECKE, Riemmanian geometry as determined by the volume of small geodesic balls, Acta Math. 142 (1979) pp P. GRIFFITHS and 1. HARRIS, Principles of Algebraic Geometry, Wiley, 1976 D. GROMOLL, W. KLINGENBERG, and W. MEYER, Riemannsche Geometrie im Grossen, Lecture Notes in Mathematics 55, Springer-Verlag, 1975 N. GROSSMAN, Geodesics on Hilbert Manifolds, Princeton University Press Notes, 1964 (referred to in McAlpin's thesis) P. GUNTHER, Huygens' Principle and Hadamard's Conjecture, Math. Intelligencer, 13 No.2 (1991) pp J. HADAMARD, Les surfaces a courbures opposees, J. Math. Pures Appl. 4 (1898) pp S. HELGASON, Some remarks on the exponential mapping for an affine connections, Math. Scand. 9 (1961) pp S. HELGASON, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, 1978 (later version of Differential Geometry and Symmetric Spaces, 1962) S. HELGASON, Wave Equations on Homogeneous Spaces, Lecture Notes in Mathematics 1077, Springer-Verlag (1984) pp N. HICKS, A theorem on affine connexions, Ill. J. Math. 3 (1959) pp H. HOPF and W. RINOW, Uber den Begriff der vollstandigen differentialgeometrischen Flache, Commentarii Math. He/v. 3 (1931) pp M.e. IRWIN, On the stable manifold theorem, Bull. London Math. Soc. (1970) pp J. KELLEY, General Topology, Van Nostrand, 1955 S. KOBAYASHI, Differential Geometry of Complex Vector Bundles, Iwanami Shoten and Princeton University Press, 1987 S. KOBAYASHI and K. NOMIZU, Foundations of Differential Geometry I, Wiley, 1963 and 1969 S. KOBAYASHI and K. NOMIZU, Foundations of Differential Geometry II, Wiley, 1969 J.L. KOSZUL, Varihes Kiihleriennes, 1st. Mat. Pura e ApI., Sao Paulo, 1957 N. KRIKORIAN, Differentiable structure on function spaces, Trans. Amer. Math. Soc. 171 (1972) pp S. LANG, Fonctions implicites et plongements Riemanniens, Seminaire Bourbaki

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5 BIBLIOGRAPHY 359 [Mo 65J [Na 56J [am 70J CPa 63J CPa 66J CPa 68J CPa 69J CPa 70J [PaS 64J [PaT 88J [Pat 7laJ [Pat 71bJ [Po 62J [Pro 61J [Pro 70J [Re 64J [Ro 68J [Sc 60J [Sm 61J [Sm 63J [Sm 64J [Sm 67J J. MOSER, On the volume element of a manifold, Trans. Amer. Math. Soc. 120 (1965) pp J. NASH, The embedding problem for Riemannian manifolds, Ann. of Math. 63 (1956) pp H. OMORI, On the group of diffeomorphisms of a compact manifold, Proc. Symp. Pure Math. Xv, AMS (1970) pp R. PALAIS, Morse theory on Hilbert maifolds, Topology 2 (1963) pp R. PALAIS, Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966) pp R. PALAIS, Foundations of Global Analysis, Benjamin, 1968 R. P ALAIS, The Morse lemma on Banach spaces, Bull. Amer. Math. Soc. 75 (1969) pp R. P ALAIS, Critical point theory and the minimax principle, Proc. Symp. Pure Math. AMS XV (1970) pp R. PALAIS and S. SMALE, A generalized Morse theory, Bull. Amer. Math. Soc. 70 (1964) pp R. P ALAIS and C. TERNG, Critical Point Theory and Submanifold Geometry, Lecture Notes Mathematics 1353, Springer-Verlag, 1988 V.K. PATODI, Curvature and the eigenforms of the Laplace operator, J. Diff. Geom. 5 (1971) pp V.K. PATODI, An analytic proof of the Riemann-Roch-Hirzebruch theorem, J. Diff. Geom. 5 (1971) pp F. POHL, Differential geometry of higher order, Topology 1 (1962) pp Proceedings of Symposia in Pure Mathematics III, Differential Geometry, AMS, 1961 Proceedings of the Conference on Global Analysis, Berkeley, Calif. 1968, AMS, 1970 G. RESTREPO, Differentiable norms in Banach spaces, Bull. Amer. Math. Soc. 70 (1964) pp J. ROBBIN, On the existence theorem for differential equations, Proc. Amer. Math. Soc. 19 (1968) pp J. SCHWARZ, On Nash's implicit functional theorem, Commun. Pure Appl. Math. 13 (1960) pp S. SMALE, Generalized Poincare's conjecture in dimensions greater than four, Ann. of Math. 74 (1964) pp S. SMALE, Stable manifolds for differential equations and diffeomorphism, Ann. Scuola Normale Sup. Pisa III Vol. XVII (1963) pp S. SMALE, Morse theory and a non-linear generalization of the Dirichlet problem, Ann. of Math. 80 No. 2 (1964) pp S. SMALE, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73, No.6 (1967) pp

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7 Index A adjoint of d 330, 332, 333 admit partitions of unity 32 almost compact support 312 alternating 142 alternating product 126, 129 analytic 336 approximate solution 71 arc length 187 atlas 20 B Banachable 4 base space 41 Bianchi identity 227 bilinear map associated with spray 99, 194. bilinear tensor 141, 214 bilinear tensor field 142 block 288 boundary 38, 315 bounded functional 300 bounded operator 4 bracket of vector fields 115 bundle along the fiber 54 C canonical 1-form and 2-form 147 canonical lifting 95 canonical spray 189 Cartan-Hadamard theorem 246 category 2 Cauchy theorem 337 Cc-functional 300 change of variables formula for integration 295 change of variables formula for sprays 100 chart 21 Christoffel symbols 208 class 0' 7, 56 closed form 135, 281 closed graph theorem 4 closed submanifold 24 cocycle condition 42 cokernel 512 commuting vector fields 120 compactly exact 322 compatible 21 complete 218 complex analytic 335 compressible 110, 180 connection 101, 102 constant curvature 238, 247 contraction 137 contraction lemma 13 convex neighborhood 214 corners 39 cotangent bundle 58, 146 covariant derivative 191, 195, 265 critical point 92, 182 curvature operator 233 curvature tensor 227

8 362 INDEX curve 87 cut-off function 196 functor 2, 57 functor of class CP 56 D d* 277 Darboux theorem 150 decomposable 127, 128 degenerate block 288 density 304 dependence on parameters 70, 74 de Rham cohomology 322 derivation 115 derivative 6 differentiable 6 differential equations 65, 158 differential form 58, 122 differential of exponential map 240 direct product 49 direct sum 59 divergence 263 divergence theorem 329 divisor 339 domain of definition 78, 84, 90, 105 dual bundle 55, 58, 146 duality of higher degree forms 273 duality of vector fields and I-forms 141 E embedding 25 energy 184, 189, 251 exact form 135, 281, 321, 322 exact sequence 50, 52 exponential map 105, 109, 209, 240, 259 exterior derivative 125, 130 F factor bundle 51 fiber 41 fiber bundle 101 fiber product 29 finite type 62 first variation 253 flow 84,89 forms 3 frame 43 Frobenius theorem 155 frontier 315 function 31 functional 36, 187, 300, 302, 345 G Gauss lemma 214,241 Gauss theorem 330 g-distance 185 geodesic 95 geodesic flow 107 geodesically complete 218 global smoothness of flow 85 gradient 144 Green's formula 332 group manifold 163 H half plane 36 Hamiltonian 145 harmonic function 332 HB-morphism 177 hermitian operator 347 Hilbert bundle 176 Hilbert group 173, 177 Hilbert space 344 Hilbert trivialization 177 Hodge conditions 279 Hodge decomposition 279 Hodge star 273 Hodge theorem 273 homomorphism 165 Hopf-Rinow theorem 219 horizontal sub bundle 103 hyperplane 36 hypersurface 212 I immersion 24 implicit mapping theorem 18 index form 250 initial condition 65, 88 inner product 343 integrable vector bundle 154, 166 integral 11 integral curve 65 integral manifold 160 integration of density 305 integration of forms 302 interior 38 isometry 187, 208 isomorphism 83

9 INDEX 363 isotopic 111 isotopy of tubular neighborhoods 111 J Jacobi differential equation 233 Jacobi field or lift 233, 269 Jacobi identity 116 Jacobian of exponential map 268 K kernel 52 kinetic energy 145, 189 Koszul's formalism 273 L Laplace operator 264, 267, 278 Laut 3 left invariant 164 length 184,212 level hypersurface 212 Levi-Civita derivative 204 lie above 199 Lie algebra 164 Lie derivative 121, 138, 140 Lie group 163 Lie subalgebra 165 Lie subgroup 165 lifting 95, 199 linear differential equation 74 Lipschitz condition 66, 284 Lipschitz constant 66 Lis 3 local coordinates 21 local flow 65 local isomorphism 13, 109 local projection 17 local representation 44, 65, 87, 96, 98, 122, 147, 189, 194, 202, 205 local smoothness 76, 78 locally closed 23 locally convex 3 locally finite 31 M manifold 21 manifolds of maps 23 manifold with boundary 36, 297 manifold without boundary 37 mean valuer theorem 11 measure associated with a form 302 measure metric 170 metric derivative 204, 208 metric isomorphism 187,208, 212 metric spray 207 minimal geodesic 217 modeled 21, 41, 142 momentum 148 morphism 2, 8, 22 Morse-Palais lemma 182 Moser's theorem 328 multilinear tensor field 59 N natural transformation 2 negligible 315 non-degenerate 182 non-singular 141, 142, 149, 182 norm 343 norm of operator 4 normal bundle 54, 109 normal chart 211 normal neighborhood 211 o one-parameter subgroup 168 operation on vector field 115 operations on vector bundles 56 operator 141, 174 orientation 261, 297 oriented chart 262 oriented volume 289 orthonormal frame 262, 266 P paracompact 131 parallel 200, 202, 208 parameter 70, 158 parametrized by arc length 187 partial derivative 8 partition of unity 32 path lifting 222 perpendicular 343 Poincare lemma 135 Poisson bracket 146 polar coordinates 216, 269 positive definite 170 projection 17 proper domain of isotopy 111 pseudo Riemannian derivative 204 pseudo Riemannian manifold 171

10 364 INDEX pseudo Riemannian metric 170 pull back 30 R reduction to Hilbert group 177 refinement 31 regular 315 related vector fields 117 representation, local, see local representation residue theorem 340 Ricci tensor 232 Riemann tensor 227 Riemannian density 306 Riemannian manifold 171 Riemannian metric 170 Riemannian volume form 262 rule mapping 10 s Sard theorem 173, 286 scalar curvature 232 scaling 230 scalloped 34 second-order differential equation 96 second-order vector field 95 second variation 252 section 3 sectional curvature 230 self dual 141 semi Riemannian 173 seminegative curvature 245 semipositive operator 348 shrinking lemma 13, 67 singular 199, 315 skew symmetric 173 spectral theorem 351 spectrum 352 sphere 212 split (injection) 16 split subspace 4 split vector bundle 62 spray 97, 104, 189, 194, 208 standard 2-form 150 star operato.:- 273 step mapping 10 Stokes' theorem for rectangles 308 Stokes' theorem on a manifold 310 Stokes' theorem with singularities 316 subbundle 50, 54 submanifold 24 submersion 25 support 31,300,321 symmetric 142, 173 symmetric bilinear form on vector bundle 171 symplectic manifold 145 T tangent bundle 49 tangent curves 88 tangent space 26 tangent subbundle 153 tangent to 0 6 tangent vector 26 Taylor expansions 257 Taylor formula 11 tensor bundle 59 tensor field 59 time dependent 65, 69 toplinear isomorphism 3 topological vector space 3 total space 41 total tubular neighborhood 108 transition map 42 transpose of dexpx 243 transversal 28 trivial vector bundle 43 trivializing covering 41 trivializable 62 tube 108 tubular map 108 tubular neighborhood 108, 178 u uniqueness theorem 68 v variation formula 252 variation of a curve 236 variation at end points 242 variation through geodesics 237 VB (vector bundle) equivalent 41 VB chart 44 VB morphism 44 vector field 87, 115 vector field along curve 201 vector subbundle 103 volume form 262, 265, 303 w wedge product 125 Whitney sum 59

bibliography [Gr 61] W. Graeub, Liesche Grupen und a n zusammenhangende Mannigfaltigkeiten,

bibliography [Gr 61] W. Graeub, Liesche Grupen und a n zusammenhangende Mannigfaltigkeiten, Bibliography [Ab 62] R. Abraham, Lectures of Smale on Di erential Topology, Columbia University, 1962 [AbM 78] R. Abraham and J. Marsden, Foundations of Mechanics, second edition, Benjamin-Cummings, 1978