《张量几何(第2版)(英文版)》是Springer数学研究生丛书之一，是一部详细讲述张量几何的教程。书中对微分几何的处理方式，以及学习广义相对论需要的数学知识使得本教程对于稍微了解单变量基本微积分和一些向量代数的知识就可以完全读懂该书的内容。《张量几何(第2版)(英文版)》用以书的形式能够提供的三维或更多维的图的形式使得内容更加形象化，重点强调数学的几何。为了表达的流畅和增强可读性，许多证明都是以练习的形式展示给读者，而非长篇的列举方程。这样，读者只能亲自进行实际计算，而不是跳过现成的例子。

这本内容丰富的教程对微分几何在相对论研究中的应用是个巨大的贡献。

Introduction

0. Fundamental Not(at)ions

 1. Sets

 2. Functions

 3. Physical Background

Ⅰ. Real Vector Spaces

 1. Spaces

Subspace geometry, components

 2. Maps

Linearity, singularity, matrices

 3. Operators

Projections, eigenvMues, determinant, trace

Ⅱ. Affine Spaces

 1. Spaces

Tangent vectors, parallelism, coordinates

 2. Combinations of Points

Midpoints, convexity

 3. Maps

Linear parts, translations, components

Ⅲ. Dual Spaces

 1. Contours, Co- and Contravariance, Dual Basis

Ⅳ. Metric Vector Spaces

 1. Metrics

Basic geometry and examples, Lorentz geometry

 2. Maps

Isometries, orthogonal projections and complements, adjoints

 3. Coordinates

Orthonormal bases

 4. Diagonalising Symmetric Operators

Principal directions, isotropy

Ⅴ. Tensors and Multilinear Forms

 1. Multilinear Forms

Tensor Products, Degree, Contraction, Raising Indices

Ⅵ. Topological Vector Spaces

 1. Continuity

Metrics: topologies, homeomorphisms

 2. Limits

Convergence and continuity

 3. The Usual Topology

Continuity in finite dimensions

 4. Compactness and Completeness

Intermediate Value Theorem, convergence, extrema

Ⅶ. Differentiation and Manifolds

 1. Differentiation

Derivative as local linear approxiamation

 2. Manifolds

Charts, maps, diffeomorphisms

 3. Bundles and Fields

Tangent and tensor bundles, metric tensors

 4. Components

Hairy Ball Theorem, transformation formulae, raising indic

 5. Curves

Parametrisation, length, integration

 6. Vector Fields and Flows

First order ordinary differential equations

 7. Lie Brackets

Commuting vector fields and flows

Ⅷ. Connections and Covariant Differentiation

 1. Curves and Tangent Vectors

Representing a vector by a curve

 2. Rolling Without Turning

Differentiation along curves in embedded manifolds

 3. Differentiating Sections

Connections horizontal vectors, Christoffel symbols

 4. Parallel Transport

Integrating a connection

 5. Torsion and Symmetry

Torsion tensor of a connection

 6. Metric Tensors and Connections

Levi-Civita connection

 7. Covariant Differentiation of Tensors

Parallel transport, Ricci's Lemma, components, constancy

Ⅸ. Geodesics

 1. Local Characterisation

Undeviating curves

 2. Geodesics from a Point

Completeness, exponential map, normal coordinates

 3. Global Characterisation

Criticality of length and energy, First Variation Formula

 4. Maxima, Minima, Uniqueness

Saddle points, mirages, Twins 'Paradox'

 5. Geodesics in Embedded Manifolds

Characterisation, examples

 6. An Example of Lie Group Geometry

2x2 matrices as a pseudo-Riemannian manifold

Ⅹ. Curvature

 1. Flat Spaces

Intrinsic description of local flatness

 2. The Curvature Tensor

Properties and Components

 3. Curved Surfaces

Ganssian curvature, Gauss-Bonnet Theorem

 4. Geodesic Deviation

Tidal effects in spacetime

 5. Sectional Curvature

Schur's Theorem, constant curvature

 6. Ricci and Einstein Tensors

Signs, geometry, Einstein manifolds, conservation equation

 7. The Weyl Tensor

Ⅺ. Special Relativity

 1. Orienting Spacetimes

Causality, particle histories

 2. Motion in Flat Spacetime

Inertial frames, momentum, rest mass, mass-energy

 3. Fields

Matter tensor, conservation

 4. Forces

No scalar potentials

 5. Gravitational Red Shift and Curvature

Measurement gives a curved metric tensor

Ⅻ. General Relativity

 1. How Geometry Governs Matter

Equivalence principle, free fall

 2. What Matter does to Geometry

Einstein's equation, shape of spacetime

 3. The Stars in Their Courses

Geometry of the solar system, Schwarzschild solution

 4. Farewell Particle

Appendix.Existence and Smoothness of Flows

 1. Completeness

 2. Two Fixed Point Theorems

 3. Sequences of Functions

 4. Integrating Vector Quantities

 5. The Main Proof

 6. Inverse Function Theorem

Bibliography

Index of Notations

Index