黎曼几何和几何分析-第6版

内容简介

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  Riemannian geometry is characterized, and research is oriented towards and shaped by concepts (geodesics, connections, curvature, …) and objectives, in particular to understand certain classes of (compact) Riemannian manifolds defined by curvature conditions (constant or positive or negative curvature, …). By way of contrast, geometric analysis is a perhaps somewhat less systematic collection of techniques, for solving extremal problems naturally arising in geometry and for investigating and characterizing their solutions. It turns out that the two fields complement each other very well; geometric analysis offers tools for solving difficult problems in geometry, and Riemannian geometry stimulates progress in geometric analysis by setting ambitious goals.  It is the aim of this book to be a systematic and comprehensive introduction to Riemannian geometry and a representative introduction to the methods of geometric analysis. It attempts a synthesis of geometric and analytic methods in the study of Riemannian manifolds.  The present work is the sixth edition of my textbook on Riemannian geometry and geometric analysis. It has developed on the basis of several graduate courses I taught at the Ruhr~University Bochum and the University of Leipzig. The main new feature of the present edition is a systematic presentation of the spectrum of the Laplace operator and its relation with the geometry of the underlying Riemannian marufold. Naturally, I have also included several smaller additions and minor corrections (for which I am grateful to several readers). Moreover, the organization of the chapters has been systematically rearranged.

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目录

1 Riemannian Manifolds1.1 Manifolds and Differentiable Manifolds1.2 Tangent Spaces1.3 Submanifolds1.4 Riemannian Metrics1.5 Existence of Geodesics on Compact Manifolds1.6 The Heat Flow and the Existence of Geodesics1.7 Existence of Geodesics on Complete ManifoldsExercises for Chapter 12 Lie Groups and Vector Bundles2.1 Vector Bundles2.2 Integral Curves of Vector Fields.Lie Algebras2.3 Lie Groups2.4 Spin StructuresExercises for Chapter 23 The Laplace Operator and Harmonic Differential Forms3.1 The Laplace Operator on Functions3.2 The Spectrum of the Laplace Operator3.3 The Laplace Operator on Forms3.4 Representing Cohomology Classes by Harmonic Forms3.5 Generalizations3.6 The Heat Flow and Harmonic FormsExercises for Chapter 34 Connections and Curvature4.1 Connections in Vector Bundles4.2 Metric Connections.The Yang—Mills Functional4.3 The Levi—Civita Connection4.4 Connections for Spin Structures and the Dirac Operator4.5 The Bochner Method4.6 Eigenvalue Estimates by the Method of Li—Yau4.7 The Geometry of Submanifolds4.8 Minimal SubmanifoldsExercises for Chapter 45 Geodesics and Jacobi Fields5.1 First and second Variation of Arc Length and Energy5.2 Jacobi Fields5.3 Conjugate Points and Distance Minimizing Geodesics5.4 Riemannian Manifolds of Constant Curvature5.5 The Rauch Comparison Theorems and Other Jacobi Field Estimates5.6 Geometric Applications of Jacobi Field Estimates5.7 Approximate Fundamental Solutions and Representation Formulas5.8 The Geometry of Manifolds of Nonpositive Sectional CurvatureExercises for Chapter 5A Short Survey on Curvature and Topology6 Symmetric Spaces and Kahler Manifolds6.1 Complex Projective Space6.2 Kahler Manifolds6.3 The Geometry of Symmetric Spaces6.4 Some Results about the Structure of Symmetric Spaces6.5 The Space Sl(n,IR)/SO(n,IR)6.6 Symmetric Spaces of Noncompact TypeExercises for Chapter 67 Morse Theory and Floer Homology7.1 Preliminaries: Aims of Morse Theory7.2 The Palais—Smale Condition,Existence of Saddle Points7.3 Local Analysis7.4 Limits of Trajectories of the Gradient Flow7.5 Floer Condition,Transversality and Z2—Cohomology7.6 Orientations and Z—homology7.7 Homotopies7.8 Graph flows7.9 Orientations7.10 The Morse Inequalities7.11 The Palais—Smale Condition and the Existence of Closed GeodesicsExercises for Chapter 78 Harmonic Maps between Riemannian Manifolds8.1 Definitions8.2 Formulas for Harmonic Maps.The Bochner Technique8.3 The Energy Integral and Weakly Harmonic Maps8.4 Higher Regularity8.5 Existence of Harmonic Maps for Nonpositive Curvature8.6 Regularity of Harmonic Maps for Nonpositive Curvature8.7 Harmonic Map Uniqueness and ApplicationsExercises for Chapter 89 Harmonic Maps from Riemann Surfaces9.1 Two—dimensional Harmonic Mappings9.2 The Existence of Harmonic Maps in Two Dimensions9.3 Regularity ResultsExercises for Chapter 910 Variational Problems from Quantum Field Theory10.1 The Ginzburg—Landau Functional10.2 The Seiberg—Witten Functional10.3 Dirac—harmonic MapsExercises for Chapter 10A Linear Elliptic Partial Differential EquationsA.1 Sobolev SpacesA.2 Linear Elliptic EquationsA.3 Linear Parabolic EquationsB Fundamental Groups and Covering SpacesBibliographyIndex

封面

黎曼几何和几何分析-第6版

书名:黎曼几何和几何分析-第6版

作者:约斯特

页数:611

定价:¥99.0

出版社:世界图书出版公司

出版日期:2015-01-01

ISBN:9787510084447

PDF电子书大小:135MB 高清扫描完整版

百度云下载:http://www.chendianrong.com/pdf

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