黎曼几何（第二版）
| 作者 |
| Petersen, Peter |
| 丛书名 |
| 国外数学名著系列（影印版） |
| 出版社 |
| 科学出版社 |
| ISBN |
| 9787030182944 |
| 简要 |
| 简介 |
| 内容简介书籍数学书籍 本书介绍黎曼几何中的重要技巧和定理，为满足那些希望专门研究黎曼几何的学生，书中还包含大量关于较深论题的背景材料。本书还介绍了**的研究问题。各种练习散布全书，帮助读者深入理解书中内容。本书是为数不多的整合了黎曼几何的几何和分析两方面内容的专著之一，适合熟悉张量和斯托克斯定理等流形理论的读者，可作为研究生一学年课程的教材。 |
| 目录 |
| Preface Chapter 1．RiemanniaMetrics 1．RiemanniaManifolds and Maps 2．Groups and RiemanniaManifolds 3．Local Representations of Metrics 4．Doubly Warped Products 5．Exercises Chapter 2．Curvature 1．Connections 2．The ConnectioiLocal Coordinates 3．Curvature 4．The Fundamental Curvature Equations 5．The Equations of RiemanniaGeometry 6．Some Tensor Concepts 7．Further Study 8．Exercises Chapter 3．Examples 1．Computational Simplifications 2．Warped Products 3．Hyperbolic Space 4．Metrics oLie Groups 5．RiemanniaSubmersions 6．Further Study 7．Exercises Chapter 4．Hypersurfaces 1．The Gauss Map 2．Existence of Hypersurfaces 3．The Gauss—Bonnet Theorem 4．Further Study 5．Exercises Chapter 5．Geodesics and Distance 1．Mixed Partials 2．Geodesics 3．The Metric Structure of a RiemanniaManifold 4．First Variatioof Energy 5．The Exponential Map 6．Why Short Geodesics Are Segments 7．Local Geometry iConstant Curvature 8．Completeness 9．Characterizatioof Segments 10．RiemanniaIsometries 11．FurtherStudy 12．Exercises Chapter 6．Sectional Curvature ComparisoⅠ 1．The ConnectioAlong Curves 2．Second Variatioof Energy 3， Nonpositive Sectional Curvature 4．Positive Curvature 5．Basic ComparisoEstimates 6．More oPositive Curvature 7．Further Study 8．Exercises Chapter 7．The Bochner Technique 1．Killing Fields 2．Hodge Theory 3．Harmonic Forms 4．Clifford MultiplicatiooForms 5．The Curvature Tensor 6．Further Study 7．Exercises Chapter 8．Symmetric Spaces and Holonomy 1．Symmetric Spaces 2．Examples of Symmetric Spaces 3．Holonomy 4．Curvature and Holonomy 5．Further Study 6．Exercises Chapter 9．Ricci Curvature Comparison 1．Volume Comparison 2．Fundamental Groups and Ricci Curvature 3．Manifolds of Nonnegative Ricci Curvature 4．Further Study 5．Exercises Chapter 10．Convergence 1．Gromov—HausdorffConvergence 2．Holder Spaces and Schauder Estimates 3．Norms and Convergence of Manifolds 4．Geometric Applications 5．Harmonic Norms and Ricci curvature 6．Further Study 7．Exercises Chapter 11．Sectional Curvature ComparisoⅡ 1．Critical Point Theory 2．Distance Comparison 3．Sphere Theorems 4．The Soul Theorem 5．Finiteness of Betti Numbers 6．Homotopy Finiteness 7．Further Study 8．Exercises Appendix．De Rham Cohomology 1．Lie Derivatives 2．Elementary Properties 3．Integratioof Forms 4．Cech Cohomology 5．De Rham Cohomology 6．Poincare Duality 7．Degree Theory 8．Further Study Bibliography Index |
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