理论工作者的高等微分几——纤维丛、射流流形和拉格朗日理论(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 manifolds
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 bundles
3．1 Connections as tangent-valued forms
3．2 Connections as jet bundle sections
3．3 Curvature and torsion
3．4 Linear and affine 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 differentialcalculus
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 Differentialoperators 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

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