本書是作者在俄羅斯、法國、南非和瑞典多年講授黎曼幾何與張量課程講義的基礎(chǔ)上整理而成。本書通俗易懂、敘述清晰。通過閱讀本書，讀者將輕松掌握應用張量、黎曼幾何的理論以及幾何化的方法求解偏微分方程，尤其是利用近似重整化群理論將大大簡化de Sitter 空間中廣義相對論方程的求解。
Nail H. Ibragimov教授為瑞典科學家，被公認為是在微分方程對稱分析方面世界上最具權(quán)威的專家之一。他發(fā)起并構(gòu)建了現(xiàn)代群分析理論和應用方面很多新的發(fā)展。
總結(jié)了利用局部黎曼幾何和李群分析求解偏微分方程的眾多有效的方法
發(fā)展了經(jīng)典方法和新方法中的分析技巧
提供了清晰易懂的表達方式、適合廣泛的讀者
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
3 Exercises
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
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