Riemannian Geometry and Geometric Analysis

Description

This established reference work continues to provide its readers with a gateway to some of the most interesting developments in contemporary geometry. It offers insight into a wide range of topics, including fundamental concepts of Riemannian geometry, such as geodesics, connections and curvature; the basic models and tools of geometric analysis, such as harmonic functions, forms, mappings, eigenvalues, the Dirac operator and the heat flow method; as well as the most important variational principles of theoretical physics, such as Yang-Mills, Ginzburg-Landau or the nonlinear sigma model of quantum field theory. The present volume connects all these topics in a systematic geometric framework. At the same time, it equips the reader with the working tools of the field and enables her or him to delve into geometric research.  The 7th edition has been systematically reorganized and updated. Almost no page has been left unchanged. It also includes new material, for instance on symplectic geometry, as well as the Bishop-Gromov volume growth theorem which elucidates the geometric role of Ricci curvature. From the reviews:“This book provides a very readable introduction to Riemannian geometry and geometric analysis... With the vast development of the mathematical subject of geometric analysis, the present textbook is most welcome.” Mathematical Reviews “For readers familiar with the basics of differential geometry and some acquaintance with modern analysis, the book is reasonably self-contained. The book succeeds very well in laying out the foundations of modern Riemannian geometry and geometric analysis. It introduces a number of key techniques and provides a representative overview of the field.” Monatshefte für Mathematik Preface Contents 1 Riemannian Manifolds 1.1 Manifolds and Differentiable Manifolds 1.2 Tangent Spaces 1.3 Submanifolds and Foliations 1.4 Riemannian Metrics 1.5 Existence of Geodesics on Compact Manifolds 1.6 The Heat Flow and the Existence of Geodesics 1.7 Existence of Geodesics on Complete Manifolds Basic Exercises for Chap.1 Further Exercises for Chap.1 2 Lie Groups and Vector Bundles 2.1 Vector Bundles 2.2 Complex and Holomorphic Vector Bundles: Almost Complex and Complex Manifolds 2.3 Integral Curves of Vector Fields: Lie Algebras 2.4 Symplectic Structures 2.5 Lie Groups 2.6 Spin Structures Basic Exercises for Chap.2 Further Exercises for Chap.2 3 The Laplace Operator and Harmonic Differential Forms 3.1 The Laplace Operator on Functions 3.2 The Spectrum of the Laplace Operator 3.3 The Laplace Operator on Forms 3.4 Representing Cohomology Classes by Harmonic Forms 3.5 The Heat Flow and Harmonic Forms Basic Exercises for Chap.3 Further Exercises for Chap.3 4 Connections and Curvature 4.1 Connections in Vector Bundles 4.2 Metric Connections. The Yang–Mills Functional 4.3 The Levi-Civita Connection 4.4 Connections for Spin Structures and the Dirac Operator 4.5 The Bochner Method 4.6 Eigenvalue Estimates by the Method of Li–Yau Basic Exercises for Chap.4 Further Exercises for Chap.4 5 Geometry of Submanifolds 5.1 The Second Fundamental Form 5.2 The Curvature of Submanifolds 5.3 The Volume of Submanifolds 5.4 Minimal Submanifolds Basic Exercises for Chap.5 Further Exercises for Chap.5 6 Geodesics and Jacobi Fields 6.1 First and Second Variation of Arc Length and Energy 6.2 Jacobi Fields 6.3 Conjugate Points and Distance Minimizing Geodesics 6.4 Riemannian Manifolds of Constant Curvature 6.5 The Rauch Comparison Theorems and Other Jacobi Field Estimates 6.6 The Hessian of the Squared Distance Function 6.7 Volume Comparison 6.8 Approximate Fundamental Solutions and Representation Formulas 6.9 The Geometry of Manifolds of Nonpositive SectionalCurvature Basic Exercises for Chap.6 Further Exercises for Chap.6 A Short Survey on Curvature and Topology 7 Symmetric Spaces and Kähler Manifolds 7.1 Complex Projective Space 7.2 Kähler Manifolds 7.3 The Geometry of Symmetric Spaces 7.4 Some Results About the Structure of Symmetric Spaces 7.5 The Space Sl (n, R)/SO (n, R) 7.6 Symmetric Spaces of Noncompact Type as Examples of Nonpositively Curved Riemannian Manifolds Basic Exercises for Chap.7 Further Exercises for Chap.7 8 Morse Theory and Floer Homology 8.1 Preliminaries: Aims of Morse Theory 8.2 Compactness: The Palais–Smale Conditionand the Existence of Saddle Points 8.3 Local Analysis: Nondegeneracy of Critical Points, Morse Lemma, Stable and Unstable Manifolds 8.4 Limits of Trajectories of the Gradient Flow 8.5 The Morse–Smale–Floer Condition: Transversality and Z2-Cohomology 8.6 Orientations and Z-homology 8.7 Homotopies 8.8 Graph Flows 8.9 Orientations 8.10 The Morse Inequalities 8.11 The Palais–Smale Condition and the Existence of Closed Geodesics Exercises for Chap.8 9 Harmonic Maps Between Riemannian Manifolds 9.1 Definitions 9.2 Formulas for Harmonic Maps: The Bochner Technique A. B. C. 9.3 Definition and Lower Semicontinuity of the Energy Integral 9.4 Higher Regularity 9.5 Harmonic Maps into Manifolds of Nonpositive Sectional Curvature: Existence 9.6 Harmonic Maps into Manifolds of Nonpositive Sectional Curvature: Regularity 9.7 Harmonic Maps into Manifolds of Nonpositive Curvature: Uniqueness and Other Properties Basic Exercises for Chap.9 Further Exercises for Chap.9 10 Harmonic Maps from Riemann Surfaces 10.1 Two-dimensional Harmonic Mappings and Holomorphic Quadratic Differentials 10.2 The Existence of Harmonic Maps in Two Dimensions 10.3 Regularity Results Basic Exercises for Chap.10 Further Exercises for Chap.10 11 Variational Problems from Quantum Field Theory 11.1 The Ginzburg–Landau Functional 11.2 The Seiberg–Witten Functional 11.3 Dirac-Harmonic Maps Exercises for Chap.11 A Linear Elliptic Partial Differential Equations A.1 Sobolev Spaces A.2 Existence and Regularity Theory for Solutions of Linear Elliptic Equations A.3 Existence and Regularity Theory for Solutions of Linear Parabolic Equations B Fundamental Groups and Covering Spaces Bibliography Index