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Preface to the Second Edition

Published online by Cambridge University Press:  12 January 2010

Isaac Chavel
Affiliation:
City College, City University of New York
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Summary

In this second edition, the first order of business has been to correct mistakes, mathematical and typographical, large and small, and clarify a number of arguments that were unclear or given short shrift the first time round. I can only hope that, in this process, and in the process of changes and additions described below, I have not introduced any new errors.

I have added some proofs of theorems, and sketches to some of the exercises, that were originally left completely to the reader in the first edition. I have added some new notes and exercises as well.

In the text itself, I have made a few changes. I added a chapter with topics from surfaces, immediately following the chapter on coverings (Chapter IV). The chapter (Chapter V) now includes the Gauss–Bonnet theorem; but, it also contains topics of current interest, showing that the Riemannian geometry of surfaces is alive and well, and is a constant testing ground, as well as a source, of new ideas. As it contained the introduction to the isoperimetric problem in Riemannian manifolds, presenting the Bol–Fiala inequalities, and the Benjamini–Cao solution of the isoperimetric problem on the paraboloid of revolution, I thought it best to follow the chapter with isoperimetric inequalities in the classical constant curvature space forms (Chapter VI).

This last chapter (Chapter VI) is a bit different from what I presented in the first edition. New proofs were given for the isoperimetric problem in Euclidean space, with the famous proof by M. Gromov, using Stokes' theorem, now appearing in my other book Isoperimetric Inequalities: Differential Geometric and Analytic Perspectives (2001).

Type
Chapter
Information
Riemannian Geometry
A Modern Introduction
, pp. xiii - xiv
Publisher: Cambridge University Press
Print publication year: 2006

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  • Preface to the Second Edition
  • Isaac Chavel, City College, City University of New York
  • Book: Riemannian Geometry
  • Online publication: 12 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511616822.001
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  • Preface to the Second Edition
  • Isaac Chavel, City College, City University of New York
  • Book: Riemannian Geometry
  • Online publication: 12 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511616822.001
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Preface to the Second Edition
  • Isaac Chavel, City College, City University of New York
  • Book: Riemannian Geometry
  • Online publication: 12 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511616822.001
Available formats
×