Synthetic geometry of manifolds

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Synthetic geometry of manifolds / Anders Kock

資料種別:: 図書
出版情報:: Cambridge : Cambridge University Press, 2010
形態:: xiii, 302 p. ; 24 cm
シリーズ名:: Cambridge tracts in mathematics ; 180 <BA00134372>
著者名:: Kock, Anders <DA04045656>
ISBN:: 9780521116732 [0521116732]
書誌ID:: BB00315920

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注記:

Includes bibliographical references and index

主題:

Geometry, Differential; Manifolds (Mathematics)

言語:

英語

目次情報:

Preface

Acknowledgements

Calculus and linear algebra / 1：

The number line R / 1.1：

The basic infinitesimal spaces / 1.2：

The KL axiom scheme / 1.3：

Calculus / 1.4：

Affine combinations of mutual neighbour points / 1.5：

Geometry of the neighbour relation / 2：

Manifolds / 2.1：

Framings and 1-forms / 2.2：

Affine connections / 2.3：

Affine connections from framings / 2.4：

Bundle connections / 2.5：

Geometric distributions / 2.6：

Jets and jet bundles / 2.7：

Infinitesimal simplicial and cubical complex of a manifold / 2.8：

Combinatorial differential forms / 3：

Simplicial, whisker, and cubical forms / 3.1：

Coboundary/exteripr derivative / 3.2：

Integration of forms / 3.3：

Uniqueness of observables / 3.4：

Wedge/cup product / 3.5：

Involutive-distnbutions and differential forms / 3.6：

Non-abelian theory of 1-forms / 3.7：

Differential forms with values in a vector bundle / 3.8：

Crossed modules and non-abelian 2-forms / 3.9：

The tangent bundle / 4：

Tangent vectors and vector fields / 4.1：

Addition of tangent vectors / 4.2：

The log-exp bijection / 4.3：

Tangent vectors as differential operators / 4.4：

Cotangents, and the cotangent bundle / 4.5：

The differential operator of a linear connection / 4.6：

Classical differential forms / 4.7：

Differential forms with values in TM?M / 4.8：

Lie bracket of vector fields / 4.9：

Further aspects of the tangent bundle / 4.10：

Groupoids / 5：

Connections in groupoids / 5.1：

Actions of groupoids on bundles / 5.3：

Lie derivative / 5.4：

Deplacements in groupoids / 5.5：

Principal bundles / 5.6：

Principal connections / 5.7：

Holonomy of connections / 5.8：

Lie theory; non-abelian covariant derivative / 6：

Associative algebras / 6.1：

Differential forms with values in groups / 6.2：

Differential forms with values in a group bundle / 6.3：

Bianchi identity in terms of covariant derivative / 6.4：

Semidirecl products; covariant derivative as curvature / 6.5：

The Lie algebra of G / 6.6：

Group-valued vs. Lie-algebra-valued forms / 6.7：

Left-invariant distributions / 6.8：

Examples of enveloping algebras and enveloping algebra bundles / 6.10：

Jets and differential operators / 7：

Linear differential operators and their symbols / 7.1：

Linear deplacements as differential operators / 7.2：

Bundle-theoretic differential operators / 7.3：

Sheaf-theoretic differential operators / 7.4：

Metric notions / 8：

Pseudo-Riemannian metrics / 8.1：

Geometry of symmetric affine connections / 8.2：

Laplacian (or isotropic) neighbours / 8.3：

The Laplace operator / 8.4：

Appendix

Category theory / A.1：

Models; sheaf semantics / A.2：

A simple topos model / A.3：

Microlinearity / A.4：

Linear algebra over local rings; Grassmannians / A.5：

Topology / A.6：

Polynomial maps / A.7：

The complex of singular cubes / A.8：

"Nullstellensatz" in multilinear algebra / A.9：

Bibliography

Index

Preface

Acknowledgements

Calculus and linear algebra / 1：

The number line R / 1.1：

The basic infinitesimal spaces / 1.2：

The KL axiom scheme / 1.3：

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