# Riemannian Geometry During the Second Half of the Twentieth Century

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*Marcel Berger*

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.

#### Reviews & Endorsements

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

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

# Table of Contents

## Riemannian Geometry During the Second Half of the Twentieth Century

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- -1. Introduction ix10 free
- 0. Riemannian Geometry up to 1950 114 free
- A. Gauss, Riemann, Christoffel and Levi-Civita 114
- B. Van Mangoldt, Hadamard, Elie Cartan and Heinz Hopf 316
- C. Synge, Myers, Preissmann: The use of geometric tools 417
- D. Hodge, harmonic forms and the Bochner technique: The use of analysis 518
- E. Allendoerfer, Weil and Chern 619
- F. Existing tools and a brief look at the new ones 821
- G. Existing examples and a brief look at the new ones 1023

- 1. Comments on the Main Topics I, II, III, IV, V under Consideration 1326
- I. Curvature and Topology 1730
- II. The Geometrical Hierarchy of Riemannian Manifolds: Space Forms 5770
- III. The Set of Riemannian Structures on a Given Compact Manifold: Is There a Best Metric? 6376
- IV. The Spectrum, the Eigenfunctions 7588
- V. Periodic Geodesies, the Geodesic Flow 8598

- TOP. Some Other Riemannian Geometry Topics of Interest 99112
- 1. Volumes 99112
- 2. Isometric embedding 107120
- 3. Holonomy groups and special metrics: Another (very restricted) Riemannian hierarchy, Kahler manifolds 108121
- 4. Cut-loci 112125
- 5. Noncompact manifolds 114127
- 6. Bundles over Riemannian manifolds 117130
- 7. Harmonic maps between Riemannian manifolds 125138
- 8. Low dimensional Riemannian geometry 126139
- 9. Some generalizations of Riemannian geometry 127140
- 10. Submanifolds 133146

- Bibliography 137150
- Additional Bibliography 171184
- Subject Index 175188
- Author Index 187200
- Back Cover Back Cover1206