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）

與量子場論不同的是，經典場論可以用嚴格的數學方式來表述，將經典場視為光滑纖維束叢的截面。本書以莫斯科國立大學（俄羅斯）理論物理系的畢業生和研究生課程為基礎。本書旨在彙編有關纖維叢、射流流形、聯絡、分次流形和拉格朗日理論的相關材料。全書共分八章，包括纖維叢的幾何結構、射流流形、纖維束聯絡、主叢幾何結構、自然叢幾何結構、分次流形幾何結構、拉格朗日理論和交換幾何主題。本書可作為廣大數學愛好者研究數學問題的參考資料。

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