文档介绍：Tensor Analysis on Manifolds in Mathematical
Physics with Applications to Relativistic Theories.
by Valter Moretti
.it/∼moretti/
Department of Mathematics,
Faculty of Science,
University of Trento
Italy
Academic year 2005-2006
1
Contents
1 Basics on differential geometry: topological and differentiable manifolds. 5
Basics of general topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
The topology of Rn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Topological Manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Differentiable Manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Local charts and atlas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Differentiable structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Differentiable functions and diffeomorphisms. . . . . . . . . . . . . . . . . 12
Some Technical Lemmata. Differentiable Partitions of Unity. . . . . . . . . . . . 13
pactness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Existence of a differentiable partition of unity. . . . . . . . . . . . . . . . 16
2 Tensor Fields in Manifolds and Associated Geometric Structures. 17
Tangent and cotangent space in a point. . . . . . . . . . . . . . . . . . . . . . . . 17
Vectors as derivations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Cotangent space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Tensor fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Lie brackets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Tangent and cotangent space manifolds. . . . . . . . . . . . . . . . . . . . . . . . 31
3 Differential mapping and Submanifolds. 34
Push forward. . . . . . . . . . . . . . . . . . . . . . .