理論工作者的高等微分幾何--纖維叢、射流流形和拉格朗日理論(英文)

Introduction

1 Geometry of fibre bundles
1.1 Fibre bundles
1.2 Vector and affine bundles
1.3 Vector fields
1.4 Exterior and tangent-valued forms

2 Jet manifblds
2.1 First order jet manifolds
2.2 Higher order jet manifolds
2.3 Differential operators and equations
2.4 Infinite order jet formalism

3 Connections on fibre bandles
3.1 Connections as tangent-valued forms
3.2 Connections as jet bundle sections
3.3 Curvature and torsion
3.4 Linear and amne connections
3.5 Flat connections
3.6 Connections on composite bundles

4 Geometry of principal bundles
4.1 Geometry of Lie groups
4.2 Bundles with structure groups
4.3 Principal bundles
4.4 Principal connections
4.5 Canonical principal connection
4.6 Gauge transformations
4.7 Geometry of associated bundles
4.8 Reduced structure

5 Geometry of natural bundles
5.1 Natural bundles
5.2 Linear world connections
5.3 Affine world connections

6 Geometry of graded manifolds
6.1 Grassmann-graded algebraic calculus
6.2 Grassmann-graded differential calculus
6.3 Graded manifolds
6.4 Graded differential forms

7 Lagrangian theory
7.1 Variational bicomplex
7.2 Lagrangian theory on fibre bundles
7.3 Grassmann-graded Lagrangian theory
7.4 Noether identities
7.5 Gauge symmetries

8 Topics on commutative geometry
8.1 Commutative algebra
8.2 Differential operators on modules
8.3 Homology and cohomology of complexes
8.4 Differential calculus over a commutative ring
8.5 Sheaf cohomology
8.6 Local-ringed spaces

Bibliography
Index
編輯手記