Universitext

Riemannian Geometry and Geometric Analysis

Authors: Jost, Jürgen

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  • Continues to lead its readers to some of the hottest topics of contemporary mathematical research
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  • ISBN 978-3-662-04672-2
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About this Textbook

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 system­ atic 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 geom­ etry stimulates progress in geometric analysis by setting ambitious goals. It is the aim of this book to be a systematic and comprehensive intro­ duction to Riemannian geometry and a representative introduction to the methods of geometric analysis. It attempts a synthesis of geometric and an­ alytic methods in the study of Riemannian manifolds. The present work is the third 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 first main new feature of the third edition is a new chapter on Morse theory and Floer homology that attempts to explain the relevant ideas and concepts in an elementary manner and with detailed examples.

Reviews

From the reviews of the first and second editions: "... a very readable introduction to Riemannian geometry and geometric analysis. The author focuses on using analytic methods in the study of some fundamental theorems in Riemannian geometry, e.g., the Hodge theorem the Rauch comparison theorem, the Lyusternik and Fet theorem and the existence of harmonic mappings. With the vast development of the mathematical subject of geometric analysis, the present textbook is most welcome. ... The book is made more interesting by the perspectives in various sections, where the author mentions the history and development of the material and provides the reader with references" Mathematical Reviews " ...the tensorial and intrinsic methods are not separated in the discussions. Instead, the easiest to handle is used for the proofs. ...this symbiosis of these methods gives a deeper understanding for clever readers. ... The book develops the subjects very systematically and takes it beyond the standard introductory topics successfully. I recommend it to everybody, who is already acquainted with Riemannian Geometry, but wants to know it better and deeper or is interested in further nice investigations. Acta Scientarium Mathematicarum

"This book provides a very readable introduction to Riemannian geometry and geometric analysis. The author focuses on using analytic methods in the study of some fundamental theorems in Riemannian geometry, e.g., the Hodge theorem, the Rauch comparison theorem, the Lyusternik and Fet theorem and the existence of harmonic mappings. With the vast development of the mathematical subject of geometric analysis, the present textbook is most welcome. It is a good introduction to Riemannian geometry. The book is made more interesting by the perspectives in various sections. where the author mentions the history and development of the material and provides the reader with references." Math. Reviews. The 2nd ed. includes new material on Ginzburg-Landau, Seibert-Witten functionals, spin geometry, Dirac operators.

From the reviews of the fifth edition:

"The text under consideration here – Riemannian Geometry And Geometric Analysis, 5th edition – is completely … a very worthy addition indeed to Jost’s textbook oeuvre. … This not only makes the presentation more contemporary than most texts, but also covers more topics closer to the frontiers of research. … the book actually ends up covering a lot more then it seems to, making it that much more impressive." (Andrew Locascio, The Mathematical Association of America, September, 2009)


Table of contents (8 chapters)

  • Foundational Material

    Jost, Jürgen

    Pages 1-78

  • De Rham Cohomology and Harmonic Differential Forms

    Jost, Jürgen

    Pages 79-99

  • Parallel Transport, Connections, and Covariant Derivatives

    Jost, Jürgen

    Pages 101-164

  • Geodesics and Jacobi Fields

    Jost, Jürgen

    Pages 165-230

  • Symmetric Spaces and Kähler Manifolds

    Jost, Jürgen

    Pages 231-279

Buy this book

eBook $69.99
price for USA (gross)
  • ISBN 978-3-662-04672-2
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
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Bibliographic Information

Bibliographic Information
Book Title
Riemannian Geometry and Geometric Analysis
Authors
Series Title
Universitext
Copyright
2002
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-662-04672-2
DOI
10.1007/978-3-662-04672-2
Series ISSN
0172-5939
Edition Number
3
Number of Pages
XIII, 535
Number of Illustrations and Tables
14 b/w illustrations
Topics