Synthetic geometry of manifolds

Başlık:

Yazar:

Kock, Anders.

ISBN:

9780521116732

Ek Yazar:

Kock, Anders.

Yayım Bilgisi:

New York : Cambridge University Press, c2010.

Fiziksel Tanım:

xiii, 302 p. : ill. ; 24 cm.

Contents:

1. Calculus and linear algebra -- 2. Geometry of the neighbour relation -- 3. Combinatorial differential forms -- 4. The tangent bundle -- 5. Groupoids -- 6. Lie theory; non-abelian covariant derivative -- 7. Jets and differential operators -- 8. Metric notions.

Konu Terimleri:

Geometri, Diferensiyel.

Manifoldlar (Matematik)

Geometry, Differential.

Manifolds (Mathematics)

Differentialgeometrie.

Mannigfaltigkeit.

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Library	Materyal Türü	Barkod	Yer Numarası	Durum
Searching... Pamukkale Merkez Kütüphanesi	Kitap	0060322	QA641K735 2010	Searching... Unknown

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This elegant book is sure to become the standard introduction to synthetic differential geometry. It deals with some classical spaces in differential geometry, namely 'prolongation spaces' or neighbourhoods of the diagonal. These spaces enable a natural description of some of the basic constructions in local differential geometry and, in fact, form an inviting gateway to differential geometry, and also to some differential-geometric notions that exist in algebraic geometry. The presentation conveys the real strength of this approach to differential geometry. Concepts are clarified, proofs are streamlined, and the focus on infinitesimal spaces motivates the discussion well. Some of the specific differential-geometric theories dealt with are connection theory (notably affine connections), geometric distributions, differential forms, jet bundles, differentiable groupoids, differential operators, Riemannian metrics, and harmonic maps. Ideal for graduate students and researchers wishing to familiarize themselves with the field.

Preface	p. ix
Acknowledgements	p. xiii
1 Calculus and linear algebra	p. 1
1.1 The number line R	p. 1
1.2 The basic infinitesimal spaces	p. 3
1.3 The KL axiom scheme	p. 11
1.4 Calculus	p. 16
1.5 Affine combinations of mutual neighbour points	p. 24
2 Geometry of the neighbour relation	p. 27
2.1 Manifolds	p. 27
2.2 Framings and 1-forms	p. 37
2.3 Affine connections	p. 42
2.4 Affine connections from framings	p. 51
2.5 Bundle connections	p. 56
2.6 Geometric distributions	p. 60
2.7 Jets and jet bundles	p. 71
2.8 Infinitesimal simplicial and cubical complex of a manifold	p. 78
3 Combinatorial differential forms	p. 81
3.1 Simplicial, whisker, and cubical forms	p. 81
3.2 Coboundary/exteripr derivative	p. 90
3.3 Integration of forms	p. 95
3.4 Uniqueness of observables	p. 104
3.5 Wedge/cup product	p. 108
3.6 Involutive-distnbutions and differential forms	p. 112
3.7 Non-abelian theory of 1-forms	p. 114
3.8 Differential forms with values in a vector bundle	p. 119
3.9 Crossed modules and non-abelian 2-forms	p. 121
4 The tangent bundle	p. 124
4.1 Tangent vectors and vector fields	p. 124
4.2 Addition of tangent vectors	p. 126
4.3 The log-exp bijection	p. 128
4.4 Tangent vectors as differential operators	p. 133
4.5 Cotangents, and the cotangent bundle	p. 135
4.6 The differential operator of a linear connection	p. 137
4.7 Classical differential forms	p. 139
4.8 Differential forms with values in TM→M	p. 144
4.9 Lie bracket of vector fields	p. 146
4.10 Further aspects of the tangent bundle	p. 150
5 Groupoids	p. 154
5.1 Groupoids	p. 154
5.2 Connections in groupoids	p. 158
5.3 Actions of groupoids on bundles	p. 164
5.4 Lie derivative	p. 169
5.5 Deplacements in groupoids	p. 171
5.6 Principal bundles	p. 175
5.7 Principal connections	p. 178
5.8 Holonomy of connections	p. 184
6 Lie theory; non-abelian covariant derivative	p. 193
6.1 Associative algebras	p. 193
6.2 Differential forms with values in groups	p. 196
6.3 Differential forms with values in a group bundle	p. 200
6.4 Bianchi identity in terms of covariant derivative	p. 204
6.5 Semidirecl products; covariant derivative as curvature	p. 206
6.6 The Lie algebra of G	p. 210
6.7 Group-valued vs. Lie-algebra-valued forms	p. 212
6.8 Infinitesimal structure of m 1 (e) $ G	p. 215
6.9 Left-invariant distributions	p. 221
6.10 Examples of enveloping algebras and enveloping algebra bundles	p. 223
7 Jets and differential operators	p. 225
7.1 Linear differential operators and their symbols	p. 225
7.2 Linear deplacements as differential operators	p. 231
7.3 Bundle-theoretic differential operators	p. 233
7.4 Sheaf-theoretic differential operators	p. 234
8 Metric notions	p. 239
8.1 Pseudo-Riemannian metrics	p. 239
8.2 Geometry of symmetric affine connections	p. 243
8.3 Laplacian (or isotropic) neighbours	p. 248
8.4 The Laplace operator	p. 254
Appendix	p. 262
A.1 Category theory	p. 261
A.2 Models; sheaf semantics	p. 263
A.3 A simple topos model	p. 269
A.4 Microlinearity	p. 272
A.5 Linear algebra over local rings; Grassmannians	p. 274
A.6 Topology	p. 279
A.7 Polynomial maps	p. 282
A.8 The complex of singular cubes	p. 285
A.9 "Nullstellensatz" in multilinear algebra	p. 291
Bibliography	p. 293
Index	p. 298

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