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© 2017

Riemannian Geometry and Geometric Analysis

Benefits

  • Continues to lead its readers to some of the hottest topics of contemporary research

  • Each chapter now includes additional basic exercises to test the reader’s understanding

  • Features new material on Ricci curvature

Textbook

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Jürgen Jost
    Pages 1-49
  3. Jürgen Jost
    Pages 51-114
  4. Jürgen Jost
    Pages 163-232
  5. Jürgen Jost
    Pages 233-249
  6. Jürgen Jost
    Pages 251-315
  7. Jürgen Jost
    Pages 317-325
  8. Jürgen Jost
    Pages 391-487
  9. Back Matter
    Pages 657-697

About this book

Introduction

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



Keywords

53B21, 53L20, 32C17, 35I60, 49-XX, 58E20, 57R15 Riemannian geometry Riemannian manifolds Lie groups symplectic geometry vector bundles Laplace operator harmonic functions harmonic maps curvature Dirac operator geometry of submanifolds geodesics Jacobi fields symmetric spaces Kähler manifolds Morse theory Floer homology quantum field theory variational problems theoretical physics variational principles

Authors and affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesMax Planck SocietyLeipzigGermany

About the authors

Jürgen Jost is Codirector of the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany, an Honorary Professor at the Department of Mathematics and Computer Sciences at Leipzig University, and an External Faculty Member of the Santa Fe Institute for the Sciences of Complexity, New Mexico, USA.

He is the author of a number of further Springer textbooks including Postmodern Analysis (1997, 2002, 2005), Compact Riemann Surfaces (1997, 2002, 2006), Partial Differential Equations (2002, 2007, 2013), Differentialgeometrie und Minimalflächen (1994, 2007, 2014, with J. Eschenburg), Dynamical Systems (2005), Mathematical Concepts (2015), as well as several research monographs, such as Geometry and Physics (2009), and many publications in scientific journals.





Bibliographic information

Reviews

“The present volume ends with two appendices (on linear elliptic partial differential equations and topological results about fundamental groups and covering spaces) and a rich bibliography of 454 items, including some classical books and papers. All the material, written in a clear and precise style, is carefully developed, many examples supporting the understanding. In the reviewer’s opinion, this is an excellent book, a very useful addition to any good library.” (Gabriel Eduard Vilcu, zbMATH 1380.53001, 2018)