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理论工作者的高等微分几何:纤维丛、射流流形和拉格朗日理论(英文)
作者:[俄]根纳迪·萨达纳什维利
出版社:哈尔滨工业大学出版社
出版时间:2021-08-01
ISBN:9787560343976
定价:¥68.00
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内容简介
《理论工作者的高等微分几何:纤维丛、射流流形和拉格朗日理论(英文)》是一部英文版的数学专著,中文书名可译为《理论工作者的高等微分几何——纤维丛、射流流形和拉格朗日理论》。 《理论工作者的高等微分几何:纤维丛、射流流形和拉格朗日理论(英文)》的作者是根纳迪·萨达纳什维利(Gennadi Sardanashvily),理论物理学家和数学物理学家,1973年毕业于莫斯科国立大学,1980年获得博士学位,1998年获得理学博士学位。莫斯科国立大学理论物理系首席研究科学家,发表了300多篇科学论文,出版了23部教科书和专著。正如《理论工作者的高等微分几何:纤维丛、射流流形和拉格朗日理论(英文)》作者在绪论中所指出:与量子场论不同的是,经典场论可以用严格的数学方式表述,将经典场视为光滑纤维丛的截面。对于R上的纤维丛,不定常的非相对论力学也是如此,《理论工作者的高等微分几何:纤维丛、射流流形和拉格朗日理论(英文)》旨在汇编有关纤维丛、射流流形、联络、分次流形和拉格朗日理论的相关材料。《理论工作者的高等微分几何:纤维丛、射流流形和拉格朗日理论(英文)》以莫斯科国立大学(俄罗斯)理论物理系的本科生和研究生课程为基础。《理论工作者的高等微分几何:纤维丛、射流流形和拉格朗日理论(英文)》适用于广大的数学家、数学物理学家和理论物理学家。它默认读者已经掌握了一些基本的微分几何知识。在《理论工作者的高等微分几何:纤维丛、射流流形和拉格朗日理论(英文)》中,所有的态射都是光滑的(即C∞类型),流形是光滑实的和有限维的。光滑实流形通常被假定为Hausdorff和第二可数的(即它的拓扑有可数的基)。因此,它是一个局部紧空间,一个可数紧子集的并,一个可分空间(即它有一个可数稠密子集),一个仿紧且完全正则的空间。在仿紧的情况下,一个光滑流形允许用光滑实函数来对整体进行分解。除非另有说明,否则假定流形是连通的(也就是说,是弧形连通的)。我们遵循无边界的流形的概念。
作者简介
暂缺《理论工作者的高等微分几何:纤维丛、射流流形和拉格朗日理论(英文)》作者简介
目录
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
编辑手记
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
编辑手记
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