张量与黎曼几何：微分方程应用 : 英文版

Preface

Part Ⅰ Tensors and Riemannian spaces

1 Preliminaries

1.1 Vectors in linear spaces

1.1.1 Three-dimensionalvectors

1.1.2 Generalcase

1.2 Index notation. Summation convention

Exercises

2 Conservation laws

2.1 Conservation laws in classical mechanics

2.1.1 Free fall of a body near the earth

2.1.2 Fall of a body in a viscous fluid

2.1.3 Discussion of Kepler's laws

2.2 General discussion of conservation laws

2.2.1 Conservationlaws for ODEs

2.2.2 Conservation laws for PDEs

2.3 Conserved vectors defined by symmetries

2.3.1 Infinitesimal symmetries of differential equations

2.3.2 Euler-Lagrange equations. Noether's theorem ...

2.3.3 Method of nonlinear self-adjointness

2.3.4 Short pulse equation

2.3.5 Linear equations

Exercises

3 Introduction of tensors and Riemannian spaces

3.1 Tensors

3.1.1 Motivation

3.1.2 Covariant and contravariant vectors

3.1.3 Tensor algebra

3.2 Riemannian spaces

3.2.1 Differential metric form

3.2.2 Geodesics. The Christoffel symbols

3.2.3 Covariant differentiation. The Riemann tensor

3.2.4 Flat spaces

3.3 Application to ODEs

Exercises

4 Motions in Riemannian spaces

4.1 Introduction

4.2 Isometric motions

4.2.1 Definition

4.2.2 Killing equations

4.2.3 Isometric motions on the plane

4.2.4 Maximal group of isometric motions

4.3 Conformal motions

4.3.l Definition

4.3.2 Generalized Killing equations

4.3.3 Conformally flat spaces

4.4 Generalized motions

4.4.l Generalized motions. their invariants and defect

4.4.2 Invariant family of spaces

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

6.1.1 Covariant form of determining equations

6.1.2 Equivalence transformations

6.1.3 Existence of conformally invariant equations

6.2 Spaces with nontrivial conformal group

6.2.1 Definition of nontrivial conformal group

6.2.2 Classification of four-dimensional spaces

6.2.3 Uniqueness theorem

6.2.4 On spaces with trivial conformal group

6.3 Standard form of second-order equations

6.3.1 Curved wave operator in V4 with nontrivial conformal group

6.3.2 Standard form of hyperbolic equations with nontrivial conformal group

……

Part Ⅲ Theory of relativity

Bibliography

Index

《张量与黎曼几何：微分方程应用（英文版）》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。《张量与黎曼几何：微分方程应用（英文版）》通俗易懂、叙述清晰。通过阅读《张量与黎曼几何：微分方程应用（英文版）》，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。