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Morita S. — Geometry of differential forms

Morita S. — Geometry of differential forms

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Название: Geometry of differential forms

Автор: Morita S.

Аннотация:

Since the times of Gauss, Riemann, and Poincaré, one of the principal goals of the study of manifolds has been to relate local analytic properties of a manifold with its global topological properties. Among the high points on this route are the Gauss-Bonnet formula, the de Rham complex, and the Hodge theorem; these results show, in particular, that the central tool in reaching the main goal of global analysis is the theory of differential forms.

The book by Morita is a comprehensive introduction to differential forms. It begins with a quick introduction to the notion of differentiable manifolds and then develops basic properties of differential forms as well as fundamental results concerning them, such as the de Rham and Frobenius theorems. The second half of the book is devoted to more advanced material, including Laplacians and harmonic forms on manifolds, the concepts of vector bundles and fiber bundles, and the theory of characteristic classes. Among the less traditional topics treated is a detailed description of the Chern-Weil theory.

The book can serve as a textbook for undergraduate students and for graduate students in geometry.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2001

Количество страниц: 321

Добавлена в каталог: 12.09.2007

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

$\epsilon$ -neighborhood      3
$\mathbb{C}$       21
$\mathbb{C}\mathrm{P}^n$       22
$\mathbb{C}^r$ function      5
$\mathbb{C}^r$ map      5
$\mathbb{C}^\infty$ atlas      15
$\mathbb{C}^\infty$ diffeomorphism      24
$\mathbb{C}^\infty$ differentiable homeomorphism      5
$\mathbb{C}^\infty$ differentiable manifold      15
$\mathbb{C}^\infty$ function      5 23
$\mathbb{C}^\infty$ manifold      15
$\mathbb{C}^\infty$ map      5 24
$\mathbb{C}^\infty$ singular cochain complex      104
$\mathbb{C}^\infty$ singular k chain      103
$\mathbb{C}^\infty$ singular k-simplex      103
$\mathbb{C}^\infty$ structure      15
$\mathbb{C}^\infty$ triangulation      101
$\mathbb{C}^\infty$ vector field      9
$\mathbb{H}^n$       44
$\mathbb{R}, \mathbb{R}^2, \mathbb{R}^3$       2
$\mathbb{R}\mathrm{P}^n$       21
$\mathbb{R}^n$       2 6
$\mathrm{P}^n$       21
$\mathrm{T}_x \mathbb{R}^n$       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
Chern class      206
Chern number      225
Chern — Simons form      287
Classes $\mathbb{C}^r$ , $\mathbb{C}^\infty$       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
GL(n; $\mathbb{C}$ )      22
Gradient      148
Graph      55
Grassmann algebra      63
Green's operator      161
GZ(n; $\mathbb{R}$ )      20
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      94
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      170
Nerve      119
Newton's formula      195
Nonzero section      172
Normal bundle      174
Null cobordant      226
O(N)      22
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
Second countability axiom      12

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