nach oben

2017 | Buch

Kapitel lesen Erstes Kapitel lesen

Riemannian Geometry and Geometric Analysis

verfasst von: Prof. Jürgen Jost

Verlag: Springer International Publishing

Buchreihe : Universitext

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

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

Inhaltsverzeichnis

Frontmatter

Chapter 1 Riemannian Manifolds

Abstract

In this chapter, the basic geometric concepts of Riemannian geometry are introduced. After treating differentiable manifolds, we introduce Riemannian metrics on them. Key concepts are the exponential map, Riemann normal coordinates, and in particular, that of a geodesic. The existence of geodesics is treated with two different methods. The first method is based on the local existence and uniqueness of geodesics. The second method is the heat flow method that gained prominence through Perelman’s solution of the Poincar conjecture by the Ricci flow method.

Jürgen Jost

Chapter 2 Lie Groups and Vector Bundles

Abstract

In this chapter, another fundamental concept is introduced, that of a vector bundle. The structure group of a vector bundle is a Lie group, and we shall therefore use this opportunity to also discuss Lie groups and their infinitesimal versions, the Lie algebras. Complex and symplectic structures are also discussed. Spin geometry is treated in detail.

Jürgen Jost

Chapter 3 The Laplace Operator and Harmonic Differential Forms

Abstract

This chapter introduces basic concepts and methods from analysis, in particular, the Laplace-Beltrami operator. The essential properties of its spectrum are shown and relationships with the underlying geometry are discussed. The discussion then turns to the operation of the Laplace operator on differential forms and de Rham cohomology groups, together with the essential tools from elliptic PDE for treating these groups. The existence of harmonic forms representing cohomology classes is proved both by a variational method and by the heat flow method. The spectrum of the Laplacian on differential forms is also discussed.

Jürgen Jost

Chapter 4 Connections and Curvature

Abstract

This chapter begins with fundamental geometric concepts: the general theory of connections. This leads to an important functional, the Yang-Mills functional. The fundamental connection of Riemannian geometry is the Levi-Civita connection, which, in particular, respects the metric. From that connection, we obtain the Riemann curvature tensor. This in turn defines the basic curvature notions of Riemannian geometry, sectional and Ricci curvature. The Weitzenböck formula identifies the difference between the Laplacian and the contracted square of the Levi-Civita connection in terms of curvature quantities. The Levi-Civita connection also induces connections on spin structures. This makes the definition of the Dirac operator possible. The Weitzenböck formula leads to the Bochner method with which the first eigenvalue of the Laplacian on manifolds of positive Ricci curvature can be estimated. Also, by this method, there are no nontrivial harmonic 1-forms on compact Riemannian manifolds of positive Ricci curvature. The chapter further describes the method of Li and Yau for obtaining eigenvalue estimates through gradient bounds for eigenfunctions.

Jürgen Jost

Chapter 5 Geometry of Submanifolds

Abstract

This chapter is devoted to the geometry of submanifolds, in particular, minimal submanifolds.

Jürgen Jost

Chapter 6 Geodesics and Jacobi Fields

Abstract

This chapter introduces Jacobi fields, proves the Rauch comparison theorems for Jacobi fields and applies these results to geodesics. Then the global geometry of spaces of nonpositive curvature is developed.

Jürgen Jost

A Short Survey on Curvature and Topology

Abstract

This section presents an account of many global results of Riemannian geometry not covered in the main text.

Jürgen Jost

Chapter 7 Symmetric Spaces and Kähler Manifolds

Abstract

This chapter treats Kähler manifolds and symmetric spaces as important examples of Riemannian manifolds in detail.

Jürgen Jost

Chapter 8 Morse Theory and Floer Homology

Abstract

This chapter introduces Morse theory and Floer homology.

Jürgen Jost

Chapter 9 Harmonic Maps Between Riemannian Manifolds

Abstract

In this chapter, harmonic maps between Riemannian manifolds are treated, utilizing the previously introduced techniques from geometric analysis. Several existence theorems are proved and applied to Riemannian geometry. The treatment uses an abstract approach based on convexity that brings out the fundamental structures.

Jürgen Jost

Chapter 10 Harmonic Maps from Riemann Surfaces

Abstract

In this chapter, harmonic maps from Riemann surfaces are discussed. We encounter here the phenomenon of conformal invariance which distinguishes the two-dimensional case from the higher dimensional one.

Jürgen Jost

Chapter 11 Variational Problems from Quantum Field Theory

Abstract

This chapter explores some deep connections between physics, geometry and analysis. It treats variational problems from quantum field theory, in particular the Ginzburg-Landau and Seiberg-Witten equations, and a mathematical version of the nonlinear supersymmetric sigma model. The background material on spin geometry and Dirac operators is already developed in earlier chapters.

Jürgen Jost

Backmatter

Titel: Riemannian Geometry and Geometric Analysis
verfasst von: Prof. Jürgen Jost
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-61860-9
Print ISBN: 978-3-319-61859-3
DOI: https://doi.org/10.1007/978-3-319-61860-9