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Index A adjoint of d 223 admit partitions of unity 32 almost compact support 205 alternating 4, 117, 132 alternating product 120 analytic 234 approximate solution 67 arc length 176 atlas 20 automorphism 3 B base space 38 bilinear form 132 bilinear map associated with spray 89 bilinear tensor 133 bilinear tensor field 133 block 184 boundary 34, 35, 208 bounded functional 196 bracket of vector fields 105 bundle along the fiber 51 C C p 7, 22, 52 canonical 1-form and 2-form 138 canonical lifting 85 canonical spray 178 category 2 Cauchy theorem 235 C c -functional 196 change of variables formula for integration 191 change of variables formula for sprays 91 chart 21 class C p 6, 52 closed form 126 closed submanifold 24 cocycle condition 39 cokernel 48 commuting vector fields 111 compact support 195 compatible 21 complex analytic 234 compressible 100, 103 connection 91, 93 contraction 12, 128 contraction lemma 13 corners 36 cotangent bundle 55, 137 cotangent vector 137 critical point 82, 170, 182 curve 61, 78 D d 223 Darboux theorem 141 decomposable 118, 119 degenerate block 184 247
248 index dependence on parameters 66 de Rham cohomology 215 derivation 106 derivative 6 determinant 225 di erentiable 6, 52 di erential equations 65, 148 di erential form 55, 113 dimension 21 direct product 55 direct sum 55 divergence 223, 225 divergence theorem 230 divisor 237 domain of definition 75, 79, 96 dual bundle 52, 137 duality of vector fields and 1-forms 134 E E-manifold 21 embedding 24 endomorphism 2 euclidean space 5 exact form 126, 214 exact sequence 46, 50 exponential map 96, 162 exterior derivative 116, 121 F factor bundle 48 fiber 38 fiber bundle 92 fiber product 28 finite type 58 flow 75, 79 forms 4 frame 40 Frobenius theorem 145 frontier 207 function 31 functional 34, 196, 198 functor 3, 52 functor of class C p 52 G Gauss theorem 231 g-distance 173 geodesic 86 geodesic flow 97 global scalar product 232 global smoothness of flow 75 gradient 135, 224 Green s formula 233 group manifold 153 H half plane 34 Hamiltonian 135 homomorphism 154 horizontal subbundle 94 hyperplane 34 I immersion 24 implicit mapping theorem 17 initial condition 61, 78 integrable vector bundle 144, 155 integral 10 integral curve 61, 78 integral manifold 150 integration of forms 198 interior 35 inverte mapping theorem 13 isometry 176 isomorphism 2, 14, 22 isotopic 101 isotopy of tubular neighborhoods 102 J Jacobian determinant 189 Jacobi identity 107 K kernel 49 kinetic energy 136, 177 L Laplace operator 223, 233 Laut 3 length 173 Lie algebra 154 Lie derivative 111, 128 Lie group 153 Lie subalgebra 155 Lie subgroup 155 lifting 85 linear di erential equation 70 Lipschitz condition 62, 180 Lipschitz constant 62 Lis 3 local coordinates 21
index 249 local flow 61 local isomorphism 13, 22 local projection 16 local representation 41, 55, 77, 89, 113, 138 local smoothness 72 locally closed 23 locally finite 31 locally isomorphic forms 140 M manifold 21 manifold with boundary 34, 208 Maut(E) 164 MB morphism 166 mean value theorem 11 measure associated with a form 197 measure 0 180 metric 159 metric bundle 165 metric isomorphism 5, 176 metric trivialization 165 modeled 21, 38, 133 momentum 139 morphism 2, 7, 22 Morsemma 172 multilinear tensor field 56 N natural transformation 3 negligible 208 non-degenerate 171 non-singular 132, 133, 140, 171, 225 norm of operator 4 normal bundle 51, 100 O one-parameter subgroup 157 operation on vector field 105 operations on vector bundles 52 operator 131, 162 orientation 193, 221, 226 oriented chart 193, 221 oriented volume 185 outward normal 231 P paracompact 31 parallel 200, 202, 208 parameter 66 parametrized by arc length 176 partial derivative 8 partition of unity 31 Poincaré lemma 126 Poisson bracket 136 positive definite 159 projection 16 proper domain of isotopy 101 pseudo Riemannian manifold 160 pseudo Riemannian metric 159, 160 pull back 29, 43 R reduction to metric group 166 refinement 31 regular 208 related vector fields 108 reparametrization 175 representation, local, see local representation residue theorem 238 Riemannian, Ri(E ) 159 Riemannian manifold 160 Riemannian metric 160 Riemannian volume form 221 ruled mapping 10 S Sard theorem 182 second-order di erential equation 86 second-order vector field 85 section 3 semi Riemannian 162 semipositive operator 162 shrinking lemma 12, 63 singular 208 skew symmetric 162 spray 88, 94 standard 2-form 141 step mapping 10 Stokes theorem for rectangles 201 Stokes theorem on a manifold 203 Stokes theorem with singularities 209 subbundle 47 submanifold 23 submersion 24 support 31, 195, 214 symmetric 4, 93, 132, 162 symmetric bilinear form on vector bundle 159 symplectic manifold 135
250 index T tangent bundle and map 46 tangent curves 78 tangent space 25 tangent subbundle 143 tangent to 0 6 tangent vector 25 Taylor formula 11 tensor bundle 55 tensor field 55 time dependent 61, 65 toplinear isomorphism 2 total space 38 transition map 39 transversal 27, 28 trivial vector bundle 40 trivializing covering 38 trivializable 58 tube 98 tubular map 98 tubular neighborhood 98, 169 U uniqueness theorem 64 unit outward normal 231 V VB (vector bundle) equivalent 38 VB chart 41 VB morphism 41 vector bundle 38 vector field 61, 77, 105 vector space 4 vector subbundle 94 volume form 199, 221 W wedge product 116 Whitney sum 55