张量几何

前言

　　This edition is essentially a reprinting of the Second Edition， with the addi-tion of two items to the Supplementary Bibliography namely　Dodson andParker： A Users Guide to Algebraic Topology， and Gray： Modern DifferentialGeometry of Curves and Surfaces.　　This latter text is very important since it contains Mathematica programsto perform all of the essential differential geometric operations on curves andsurfaces in 3-dimensional Euclidean space. The programs are available byanonymous ftp from bianchi.umd.edu/pub/ and are being used as supportfor a course at， among other places，

书籍目录

Introduction0. Fundamental Not(at)ions　1. Sets　2. Functions　3. Physical BackgroundI. Real Vector Spaces　1. Spaces　　Subspace geometry, components　2. Maps　　Linearity, singularity, matrices　3. Operators　　Projections, eigenvMues, determinant, traceII. Affine Spaces　1. Spaces　　Tangent vectors, parallelism, coordinates　2. Combinations of Points　　Midpoints, convexity　3. Maps　　Linear parts, translations, componentsIII. Dual Spaces　1. Contours, Co- and Contravariance, Dual BasisIV. 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, isotropyV. Tensors and Multilinear Forms　1. Multilinear Forms　　Tensor Products, Degree, Contraction, Raising IndicesVii 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, extremaVII. 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 flowsVIII. 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 connectionIX. GeodesicsX. CurvatureXI. Special RelativityXII. General RelativityBibliographyIndex of NotationsIndex

作者简介

《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。

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