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Riemannian Geometry During the Second Half of the Twentieth Century
Marcel Berger, Institut des Hautes Études Scientifiques, Bures-Sur-Yvette, France
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University Lecture Series
2000; 182 pp; softcover
Volume: 17
Reprint/Revision History:
reprinted with corrections 2002
ISBN-10: 0-8218-2052-4
ISBN-13: 978-0-8218-2052-0
List Price: US$40
Member Price: US$32
Order Code: ULECT/17
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Additional related information:
http://www.teubner.de

During its first hundred years, Riemannian geometry enjoyed steady, but undistinguished growth as a field of mathematics. In the last fifty years of the twentieth century, however, it has exploded with activity. Berger marks the start of this period with Rauch's pioneering paper of 1951, which contains the first real pinching theorem and an amazing leap in the depth of the connection between geometry and topology. Since then, the field has become so rich that it is almost impossible for the uninitiated to find their way through it. Textbooks on the subject invariably must choose a particular approach, thus narrowing the path. In this book, Berger provides a truly remarkable survey of the main developments in Riemannian geometry in the last fifty years.

One of the most powerful features of Riemannian manifolds is that they have invariants of (at least) three different kinds. There are the geometric invariants: topology, the metric, various notions of curvature, and relationships among these. There are analytic invariants: eigenvalues of the Laplacian, wave equations, Schrödinger equations. There are the invariants that come from Hamiltonian mechanics: geodesic flow, ergodic properties, periodic geodesics. Finally, there are important results relating different types of invariants. To keep the size of this survey manageable, Berger focuses on five areas of Riemannian geometry: Curvature and topology; the construction of and the classification of space forms; distinguished metrics, especially Einstein metrics; eigenvalues and eigenfunctions of the Laplacian; the study of periodic geodesics and the geodesic flow. Other topics are treated in less detail in a separate section.

While Berger's survey is not intended for the complete beginner (one should already be familiar with notions of curvature and geodesics), he provides a detailed map to the major developments of Riemannian geometry from 1950 to 1999. Important threads are highlighted, with brief descriptions of the results that make up that thread. This supremely scholarly account is remarkable for its careful citations and voluminous bibliography. If you wish to learn about the results that have defined Riemannian geometry in the last half century, start with this book.

Reprint arranged with the approval of the publisher B. G. Teubner (Stuttgart and Leipzig, Germany).

Readership

Graduate students and research mathematicians interested in differential geometry.

Reviews

"This is quite an amazing book ... The coverage ... is quite astonishing, both in breadth and in depth ... The reader is left with the feeling that essentially all the topics covered are still the subject of active research ... Another outstanding feature of the book is its extensive cross-referencing ... this feature ... made the book particularly valuable to my students in a graduate Riemannian geometry course ... they could quickly find out more about a topic which was completely new to them."

-- Bulletin of the LMS

From a review of the original edition:

"In this survey, Marcel Berger, who is among the most celebrated geometers of our time, sought to trace the development of Riemannian geometry during the second half of the twentieth century. He did it not by the vain attempt of writing an encyclopedia on the subject, but by pointing out the essential concepts, viewpoints, innovative ideas and techniques which all together lead to important results. The last section focuses on volumes, isometric embedding, holonomy groups, cut-loci, harmonic maps, submanifolds and low-dimensional Riemannian geometry. The article is masterfully written and delightful to read. In addition to the numerous digressions for newly introduced concepts, the author adds to the value of the survey by providing fertile opinions, some of them his, others those of his close colleagues and of M. Gromov in particular. The wonderful effort of the author is shown partially by the long bibliography of thirty pages, with references updated right to the very end of the century. A person who wants to learn more about Riemannian geometry will certainly do him/herself a good service by reading Berger's work."

-- Mathematical Reviews

Table of Contents

  • Additional bibliography
  • Riemannian geometry up to 1950
  • Comments on the main topics I, II, III, IV, V under consideration
  • Curvature and topology
  • The geometrical hierarchy of Riemann manifolds: Space forms
  • The set of Riemannian structures on a given compact manifold: Is there a best metric?
  • The spectrum, the eigenfunctions
  • Periodic geodesics, the geodesic flow
  • Some other Riemannian geometric topics of interest
  • Bibliography
  • Subject and notation index
  • Name index
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