Universitext

Riemannian Geometry

Authors: Gallot, Sylvestre, Hulin, Dominique, Lafontaine, Jacques

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About this Textbook

In this second edition, the main additions are a section devoted to surfaces with constant negative curvature, and an introduction to conformal geometry. Also, we present a -soft-proof of the Paul Levy-Gromov isoperimetric inequal­ ity, kindly communicated by G. Besson. Several people helped us to find bugs in the. first edition. They are not responsible for the persisting ones! Among them, we particularly thank Pierre Arnoux and Stefano Marchiafava. We are also indebted to Marc Troyanov for valuable comments and sugges­ tions. INTRODUCTION This book is an outgrowth of graduate lectures given by two of us in Paris. We assume that the reader has already heard a little about differential manifolds. At some very precise points, we also use the basic vocabulary of representation theory, or some elementary notions about homotopy. Now and then, some remarks and comments use more elaborate theories. Such passages are inserted between *. In most textbooks about Riemannian geometry, the starting point is the local theory of embedded surfaces. Here we begin directly with the so-called "abstract" manifolds. To illustrate our point of view, a series of examples is developed each time a new definition or theorem occurs. Thus, the reader will meet a detailed recurrent study of spheres, tori, real and complex projective spaces, and compact Lie groups equipped with bi-invariant metrics. Notice that all these examples, although very common, are not so easy to realize (except the first) as Riemannian submanifolds of Euclidean spaces.

Reviews

From the reviews of the third edition:

"This new edition maintains the clear written style of the original, including many illustrations … examples and exercises (most with solutions)." (Joseph E. Borzellino, Mathematical Reviews, 2005)

"This book based on graduate course on Riemannian geometry … covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features of the subject. Classical results … are treated in detail. … contains numerous exercises with full solutions and a series of detailed examples which are picked up repeatedly to illustrate each new definition or property introduced. For this third edition, some topics … have been added and worked out in the same spirit." (L'ENSEIGNEMENT MATHEMATIQUE, Vol. 50, (3-4), 2004)

"This book is based on a graduate course on Riemannian geometry and analysis on manifolds that was held in Paris. … Classical results on the relations between curvature and topology are treated in detail. The book is almost self-contained, assuming in general only basic calculus. It contains nontrivial exercises with full solutions at the end. Properties are always illustrated by many detailed examples." (EMS Newsletter, December 2005)

"The guiding line of this by now classic introduction to Riemannian geometry is an in-depth study of each newly introduced concept on the basis of a number of reoccurring well-chosen examples … . The book continues to be an excellent choice for an introduction to the central ideas of Riemannian geometry." (M. Kunzinger, Monatshefte für Mathematik, Vol. 147 (1), 2006)


Table of contents (5 chapters)

  • Differential Manifolds

    Gallot, Sylvestre (et al.)

    Pages 1-50

  • Riemannian Metrics

    Gallot, Sylvestre (et al.)

    Pages 51-105

  • Curvature

    Gallot, Sylvestre (et al.)

    Pages 106-179

  • Analysis on Manifolds and the Ricci Curvature

    Gallot, Sylvestre (et al.)

    Pages 180-215

  • Riemannian Submanifolds

    Gallot, Sylvestre (et al.)

    Pages 216-231

Buy this book

eBook $74.99
price for USA in USD (gross)
  • ISBN 978-3-642-97242-3
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $99.00
price for USA in USD
  • ISBN 978-3-540-52401-4
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Riemannian Geometry
Authors
Series Title
Universitext
Copyright
1990
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-97242-3
DOI
10.1007/978-3-642-97242-3
Softcover ISBN
978-3-540-52401-4
Series ISSN
0172-5939
Edition Number
2
Number of Pages
XIII, 286
Topics