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© 1990

Riemannian Geometry

Textbook

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages I-XIII
  2. Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine
    Pages 1-50
  3. Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine
    Pages 51-105
  4. Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine
    Pages 106-179
  5. Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine
    Pages 180-215
  6. Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine
    Pages 216-231
  7. Back Matter
    Pages 232-286

About this book

Introduction

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.

Keywords

Minimal surface Riemannian geometry Riemannian goemetry covariant derivative curvature manifold relativity

Authors and affiliations

  1. 1.Ecole Polytechnique, Unité de Recherche Associée du CNRS D 0169Centre de MathématiquesPalaiseau CedexFrance
  2. 2.Départment de Mathématiques, GETODIM - Unité de Recherche Associée du CNRS 1407Université de MontpellierMontpellier Cedex 5France

Bibliographic information

  • Book Title Riemannian Geometry
  • Authors Sylvestre Gallot
    Dominique Hulin
    Jacques Lafontaine
  • Series Title Universitext
  • DOI https://doi.org/10.1007/978-3-642-97242-3
  • Copyright Information Springer-Verlag Berlin Heidelberg 1990
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Softcover ISBN 978-3-540-52401-4
  • eBook ISBN 978-3-642-97242-3
  • Series ISSN 0172-5939
  • Series E-ISSN 2191-6675
  • Edition Number 2
  • Number of Pages XIII, 286
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Differential Geometry
    Manifolds and Cell Complexes (incl. Diff.Topology)
  • Buy this book on publisher's site

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)