Riemannian geometry and geometric analysis
type: Item
instance of: scholarly article
title: Riemannian geometry and geometric analysis
zbMATH Open document ID: 1143.53001
DOI: 10.1007/978-3-540-77341-2
author: Jürgen Jost
published in: Universitext
publication date: 2008-02-11T00:00:00Z
review text: The first four chapters of the book deal with the following basic geometric notions: differentiable manifold, tangent space, vector bundle, vector field, one-parameter group of diffeomorphisms, Lie algebras and Lie groups, Riemannian metrics, geodesics (Chapter 1); de Rham cohomology groups, harmonic forms, heat flow method (Chapter 2); general theory of connections and curvature (Chapter 3); Jacobi fields, conjugate points, Rauch comparison theorems, spaces of nonpositive sectional curvature (Chapter 4). In Chapter 5, symmetric spaces and Kähler manifolds are analyzed in detail. Chapter 6 is devoted to Morse theory and Floer homology. Chapter 7 deals with harmonic maps between Riemanian manifolds, while the case of harmonic maps from Riemannian surfaces is studied in Chapter 8. The last chapter treats variational problems from quantum field theory, in particular the Ginzburg-Landau, Seibert-Witten equations, and a mathematical version of the nonlinear supersymmetric sigma model. In two final appendices results concerning linear elliptic partial differential equations, fundamental groups, and covering spaces are included. With few exceptions, the material of this book is self-contained. Each chapter ends with a set of exercises. Most of the paragraphs have a section ``Perspectives'', written with the aim to place the material in a broader context and explain further results and directions. The main new features of this fifth edition are the systematic inclusion of flow equations and a mathematical treatment of the nonlinear sigma model of quantum field theory. For previous editions of this book see Zbl 0828.53002, Zbl 0997.53500, Zbl 1034.53001 and Zbl 1083.53001.
reviewed by: Gheorghe Pitiş
Mathematics Subject Classification ID: 53-01
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53B21
Mathematics Subject Classification ID: 58-01
zbMATH DE Number: 5233828
zbMATH Keywords: differentiable manifold
zbMATH Keywords: submanifold
zbMATH Keywords: Riemannian metric
zbMATH Keywords: geodesic
zbMATH Keywords: heat flow
zbMATH Keywords: Lie group
zbMATH Keywords: Lie algebra
zbMATH Keywords: Laplace operator
zbMATH Keywords: harmonic forms
zbMATH Keywords: cohomology classes
zbMATH Keywords: vector bundle
zbMATH Keywords: connection
zbMATH Keywords: spin structure
zbMATH Keywords: Bochner method
zbMATH Keywords: Jacobi fields
zbMATH Keywords: sectional curvature
zbMATH Keywords: symmetric spaces
zbMATH Keywords: Kähler manifold
zbMATH Keywords: Morse theory
zbMATH Keywords: Floer homology
zbMATH Keywords: harmonic map
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-540-77341-2
OpenAlex ID: W4212810988
description: scientific article; zbMATH DE number 5233828