 function 5
map 5
 atlas 15
differentiable homeomorphism 5
differentiable manifold 15
difleomorphism 24
function 5 23
manifold 15
map 5 24
singular cochain complex 104
singular fc-simplex 103
singular k chain 103
structure 15
triangulation 101
vector field 9
 22
 20
 22
 21
 23
 6
 -neighborhood 3
 21
 22
 44
 2
 21
 2 6
Abstract simplicial complex 97
Action of a group 50
Adjoint operator 154
Admissible 234
Alexander — Whitney map 133
Algebra 24 57
Alternating 63
Alternating form 63
Anti-derivation 73
Associated bundle 236
Atlas 14
Automorphism group 50
Base space 171
Basic element (in a Weil algebra) 276
Betti number 116
Bianchi’s identity 196
Boundary 45
Boundary cycle 98
Boundary operator 98
Bracket 39
Bundle map 232 235
Cartan formula 74
Cartan — Eilenberg theorem 138
Cech cohomology 119
cell 96
Chain complex 98
Characteristic class 198 238
Characteristic number 226
Chera class 206
Chern number 225
Chern — Simons form 287
Classes , 5
Classifying space 239
Closed form 60 111
Closed manifold 46
Cobordant 226
Coboundary 99
Cochain complex 98
Cocycle 99
Cocycle condition 171 233
Coherent orientation 46
Cohomologous 99
Cohomology 98
Commutative vector fields 82
Compact 27
Compatible (with a metric) 199
Complement of a knot 20
complete 43
Completely integrable 80
Complex Lie group 22
Complex manifold 21
Complex projective space 22
Complex vector bundle 171
Complexification 175
Conjugacy 290
Conjugate bundle 209
Connection 185
Connection for a complex vector bundle 205
Connection form 185 264
Connection in a general bundle 258
Connection in a principal bundle 260
Contractible 119
Contractible open covering 121
Coordinate change 15
Coordinate functions 12
Coordinate neighborhood 12
Cotangent bundle 67 177
Cotangent space 67
Covariant derivative 181 185
Covariant exterior differential 193
Covering 27
Covering manifold 51
Covering map 51
Curvature form 186 188 252 264
CYCLE 98
De Rham cohomology 111
de Rham cohomology, algebra 113
de Rham cohomology, group 112
de Rham complex 112
de Rham theorem 114
de Rham theorem for triangulated manifolds 115
de Rham theorem, concerning the product 131
Derivation 38
Diff M 43
Diffeomorphism 5 24
Diffeomorphism group 43
Differentiable manifolds 1
Differential 33
Differential form 58
Differential form, coordinate-independent definition 63
Differential ideal 87
Directional derivative 8
Discrete group 50
Distance 3
distribution 80 258
Divergence 152
Double complex 123
Dual bundle 176
Dual space 63
Elliptic PDE 161
Embedding 34
Euclidean simplicial complex 96
Euclidean space 147
Euler characteristic 164
Euler class 212 246 254
Euler form 213
Euler number 164
Euler — Poincare characteristic 164
Exact form 60 111
Existence and uniqueness of the solution of ODEs 41
Existence of partitions of unity 29
exp tX 43
Exterior algebra 58 63
Exterior differentiation 59 70
Exterior power bundle 177
Exterior product 59 69
Face 96
Fiat G bundle 288
Fiber 171 232
Fiber bundle 232
Flat connection 288
Frame field 172
| free 50
Frobenius theorem 81 88
Fundamental class 103
Fundamental vector field 119
G-structure 234
Gauss — Bonnet theorem 216
Gaussian plane 22
General linear group 20
General position 96
Geodesic 180
Gradient 148
Graph 55
Grassrnann algebra 63
Green’s operator 161
Harmonic form 155
Harmonic function 155
Hausdorff separation axiom 11
Hausdorff space 11
Hirzebruch signature theorem 227
Hodge decomposition 160
Hodge operator * 150
Hodge theorem 159
Holomorphic mapping 22
Holonomy homomorphism 289
Homeomorphism 4
Homogeneous coordinate 21
Homologous 98
Homology group 97
Homology theory of cell complexes 96
Homology theory of simplicial complexes 96
Homotopy invariance of de Rham cohomology 119
Homotopy type 119
Hopf index theorem 256
Hopf invariant 134
Hopf line bundle 174
Hopf map 25 34
Horizontal lift 259
Horizontal vector 259
Hyperbolic space 147
Immersion 34
INDEX 256
Induced bundle 236
Induced connection 198
Integrability condition 88
Integral curve 39
Integral manifold 80
Interior product 73
Intersection form 166
Intersection number 165
Invariant polynomial function 194
Inverse function theorem 5 6
Involutive 81
Isolated singular point 255
Isomorphic bundles 172 235
Jacobi identity 39
Jacobian 5
Jacobian matrix 5
k cochain 120
k-form 58
Knot 20
Kronecker product 99
Kuenneth formula 168
l-chain 97
l-simplex 96
Laplace — Bertrami operator 155
Laplacian 155
Left-hand system 47
Lens space 53
Levi — Civita connection 201
Lie algebra 39 90
Lie derivative 77
Lie group 22
Lift 259
Line bundle 171
Link with two components 140
Linking number 141
Local chart 12
Local coordinate system 1 12
Local coordinate system, positive 49
Locally finite 27
Manifold with boundary 45
Mapping degree 139
Massey products 136
Massey products, triple 136
Maurer — Cartan equation 92
Maurer — Cartan form 91
Maximal atlas 15
Maximal integral curve 41
Metric connection 199
Metric space 3
Multilinear 63
n-dimensional numerical space 2
N-dimensional sphere 17
n-dimensional torus 17
n-dimensional vector space 6
n-sphere 17
Nerve 119
Newton’s formula 195
Nonzero section 172
Normal bundle 174
Null cobordant 226
One parameter group of local transformations 42
One parameter group of transformations 43
Open covering 27
Open neighborhood 3
Open set 3
Open simplex 119
Open star 119
Open submanifold 20
Orbit 50
Orbit space 50
Ordered basis 47
Orientable 46 48 211
Orientation 47 97
Orientation preserving 49
Oriented manifold 48
Orthogonal group 22
Paracompact 27
Parallel along a curve 183
Parallel displacement 183
Partition of unity 29
Pfaffian 212
Poincare disk 167
Poincare duality 163
Poincare lemma 118
Polar coordinates 16
Polyhedron 97
Pontrjagin class 200
Pontrjagin form 201
Pontrjagin number 225
Primary obstruction 255
Principal bundle 236
Principal G-bundle 236
Product bundle 171
Product manifold 16
Projection 171
Proof of the de Rham theorem 126
Properly discontinuous 50
Pullback 72 236
Quotient bundle 174
Quotient space 50
Real projective space 21
Reducible 235
refinement 27
Regular submanifold 21
Restriction of a bundle 173
Riemannian manifold 146
Riemannian metric 146
Riemannian metric in a vector bundle 175
Right-hand system 47
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