微分流形與黎曼幾何引論

內(nèi)容概要

這是一本非常好的微分流形入門書。全書從一些基本的微積分知識入手，然后一點點深入介紹，主要內(nèi)容有：流形介紹、多變量函數(shù)和映射、微分流形和子流形、流形上的向量場、張量和流形上的張量場、流形上的積分法、黎曼流形上的微分法以及曲率。書后有難度適中的習(xí)題，全書配有很多精美的插圖。     本書非常適合初學(xué)者閱讀，可作為數(shù)學(xué)系、物理系、機械系等理工科高年級本科生和研究生的教材。作者簡介：    William M.Boothby華盛頓大學(xué)圣路易斯分校數(shù)學(xué)系榮休教授。于1949年在密歇根大學(xué)獲得博士學(xué)位，師出拓撲學(xué)大師、沃爾夫獎得主Hassler Whitney門下。除在華盛頓大學(xué)任教40余年外，他還在世界各地講授微分流形、深受學(xué)生愛戴。

書籍目錄

Ⅰ. Introduction to Manifolds  1.Preliminary Comments on Rn  2.Rn and Euclidean Space  3.Topological Manifolds  4.Further Examples of Manifolds. Cutting and Pasting  5.Abstract Manifolds. Some ExamplesⅡ. Functions of Several Variables and Mappings  1.Differentiability for Functions of Several Variables  2.Differentiability of Mappings and Jacobians  3.The Space of Tangent Vectors at a Point of Rn  4.Another Definition of Ta(Rn)  5.Vector Fields on Open Subsets of Rn  6.The Inverse Function Theorem  7.The Rank of a MappingⅢ. Differentiable Manifolds and Submanifolds  1.The Definition of a Differentiable Manifold  2.Further Examples  3.Differentiable Functions and Mappings  4.Rank of a Mapping, Immersions  5.Submanifolds  6.Lie Groups  7.The Action of a Lie Group on a Manifold. Transformation Groups  8.The Action of a Discrete Group on a Manifold  9.Covering ManifoldsⅣ. Vector Fields on a Manifold  1.The Tangent Space at a Point of a Manifold  2.Vector Fields  3.One-Parameter and Local One-Parameter Groups Acting on a Manifold  4.The Existence Theorem for Ordinary Differential Equations  5.Some Examples of One-Parameter Groups Acting on a Manifold  6.One-Parameter Subgroups of Lie Groups  7.The Lie Algebra of Vector Fields on a Manifold  8.Frobenius's Theorem  9.Homogeneous SpacesⅤ. Tensors and Tensor Fields on Manifolds  1.Tangent Covectors  2.Bilinear Forms. The Riemannian Metric  3.Riemannian Manifolds as Metric Spaces  4.Partitions of Unity  5.Tensor Fields  6.Multiplication of Tensors  7.Orientation of Manifolds and the Volume Element  8.Exterior DifferentiationⅥ. Integration on ManifoldsⅦ. Differentiation on Riemannian ManifoldsⅧ. CurvatureREFERENCESINDEX

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