Part Ⅰ Tensors and Riemannian spaces

1 Preliminaries

1.1 Vectors in linear spaces

1.1.1 Three-dimensional vectors

1.1.2 General case

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 ofa body in a viscous fluid

2.1.3 Discussion of Kepler's laws

2.2 General discussion of conservation laws

2.2.1 Conservation laws 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

4.3.1 Definition

4.3.2 Generalized Killing equations

4.3.3 Conformally flat spaces

4.4 Generalized motions

4.4.1 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

Exercises

7 Solution of the initial value problem

7.1 The Cauchy problem

7.1.1 Reduction to a particular Cauchy problem

7.1.2 Fourier transform and solution of the particular Cauchy problem

7.1.3 Simplification of the solution

7.1.4 Verification of the solution

7.1.5 Comparison with Poisson's formula

7.1.6 Solution of the general Cauchy problem

7.2 Geodesics in spaces with nontrivial conformal group

7.2.1 Outline of the approach

7.2.2 Equations of geodesics in spaces with nontrivial conformal group

7.2.3 Solution of equations for geodesics

7.2.4 Computation of the geodesic distance

7.3 The Huygens principle

7.3.1 Huygens'principle for classical wave equation

7.3.2 Huygens'principle for the curved wave operator in V4 with nontrivial conformal group

7.3.3 On spaces with trivial conformal group

Exercises

Part Ⅲ Theory of relativity

8 Brief introduction to relativity

8.1 Special relativity

8.1.1 Space-time intervals

8.1.2 The Lorentz group

8.1.3 Relativistic principle of least action

8.1.4 Relativistic Lagrangian

8.1.5 Conservation laws in relativistic mechanics

8.2 The Maxwell equations

8.2.1 Introduction

8.2.2 Symmetries of Maxwell's equations

8.2.3 General discussion of conservation laws

8.2.4 Evolutionary part of Maxwell's equations

8.2.5 Conservation laws of Eqs.(8.2.1 )and(8.2.2 )

8.3 The Dirac equation

8.3.1 Lagrangian obtained from the formal Lagrangian

8.3.2 Symmetries

8.3.3 Conservation laws

8.4 General relativity

8.4.1 The Einstein equations

8.4.2 The Schwarzschild space

8.4.3 Discussion of Mercury's parallax

8.4.4 Solutions based on generalized motions

Exercises

9 Relativity in de Sitter space

9.1 The de Sitter space

9.1.1 Introduction

9.1.2 Reminder of the notation

9.1.3 Spaces of constant Riemannian curvature

9.1.4 Killing vectors in spaces of constant curvature

9.1.5 Spaces with positive definite metric

9.1.6 Geometric realization of the de Sitter metric

9.2 The de Sitter group

9.2.1 Generators of the de Sitter group

9.2.2 Conformal transformations in R3

9.2.3 Inversion

9.2.4 Generalized translation in direction of x-axis

9.3 Approximate de Sitter group

9.3.1 Approximate groups

9.3.2 Simple method of solution of Killing's equations

9.3.3 Approximate representation of de Sitter group

9.4 Motion of a particle in de Sitter space

9.4.1 Introduction

9.4.2 Conservation laws in Minkowski space

9.4.3 Conservation laws in de Sitter space

9.4.4 Kepler's problem in de Sitter space

9.5 Curved wave operator

9.6 Neutrinos in de Sitter space

9.6.1 Two approximate representations of Dirac's equations in de Sitter space

9.6.2 Splitting of neutrinos by curvature

Exercises

Bibliography

Index