《张量与黎曼几何(微分方程应用英文版)(精)/非线性物理科学》是作者在俄罗斯、法国、 南非和瑞典多年讲授黎曼几何与张量课程讲义的基础 上整理而成。本书通俗易懂、叙述清晰。通过阅读本 书，读者将轻松掌握应用张量、黎曼几何的理论以及 几何化的方法求解偏微分方程，尤其是利用近似重整 化群理论将大大简化de Sitter 空间中广义相对论方 程的求解。
Nail H. Ibragimov教授为瑞典科学家，被公认为是在微分方程对称分析方面世界上最具权威的专家 之一。他发起并构建了现代群分析理论和应用方面很多新的发展。
- 目录
- 前辅文
- Part I Tensors and Riemannian spaces
- 1 Preliminaries
- 1.1 Vectors in linear spaces
- 1.2 Index notationSummation convention
- Exercises
- 2 Conservation laws
- 2.1 Conservation laws in classical mechanics
- 2.2 General discussion of conservation laws
- 2.3 Conserved vectors defined by symmetries
- Exercises
- 3 Introduction of tensors and Riemannian spaces
- 3.1 Tensors
- 3.2 Riemannian spaces
- 3.3 Application to ODEs
- Exercises
- 4 Motions in Riemannian spaces
- 4.1 Introduction
- 4.2 Isometric motions
- 4.3 Conformal motions
- 4.4 Generalized motions
- Exercises
- Part II Riemannian spaces of second-order equations
- 5 Riemannian spaces associated with linear PDEs
- 5.1 Covariant form of second-order equations
- 5.2 Conformally invariant equations
- Exercises
- 6 Geometry of linear hyperbolic equations
- 6.1 Generalities
- 6.2 Spaces with nontrivial conformal group
- 6.3 Standard form of second-order equations
- Exercises
- 7 Solution of the initial value problem
- 7.1 The Cauchy problem
- 7.2 Geodesics in spaces with nontrivial conformal group
- 7.3 The Huygens principle
- Exercises
- Part III Theory of relativity
- 8 Brief introduction to relativity
- 8.1 Special relativity
- 8.2 The Maxwell equations
- 8.3 The Dirac equation
- 8.4 General relativity
- Exercises
- 9 Relativity in de Sitter space
- 9.1 The de Sitter space
- 9.2 The de Sitter group
- 9.3 Approximate de Sitter group.
- 9.4 Motion of a particle in de Sitter space
- 9.5 Curved wave operator.
- 9.6 Neutrinos in de Sitter space
- Exercises
- Bibliography
- Index
