A Panoramic View of Riemannian Geometry

1. Front Matter

   Pages I-XXIII

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2. Old and New Euclidean Geometry and Analysis

   • Marcel Berger

   Pages 1-99

3. Transition: The Need for a More General Framework

   • Marcel Berger

   Pages 101-104

4. Surfaces from Gauß to Today

   • Marcel Berger

   Pages 105-142

5. Riemann’s Blueprints for Architecture in Myriad Dimensions

   • Marcel Berger

   Pages 143-218

6. A One Page Panorama

   • Marcel Berger

   Pages 219-219

7. Riemannian Manifolds as Metric Spaces and the Geometric Meaning of Sectional and Ricci Curvature

   • Marcel Berger

   Pages 221-297

8. Volumes and Inequalities on Volumes of Cycles

   • Marcel Berger

   Pages 299-368

9. Transition: The Next Two Chapters

   • Marcel Berger

   Pages 369-372

10. Riemannian Manifolds as Quantum Mechanical Worlds: The Spectrum and Eigenfunctions of the Laplacian

    • Marcel Berger

    Pages 373-429

11. Riemannian Manifolds as Dynamical Systems: the Geodesic Flow and Periodic Geodesics

    • Marcel Berger

    Pages 431-497

12. What is the Best Riemannian Metric on a Compact Manifold?

    • Marcel Berger

    Pages 499-541

13. From Curvature to Topology

    • Marcel Berger

    Pages 543-635

14. Holonomy Groups and Kähler Manifolds

    • Marcel Berger

    Pages 637-657

15. Some Other Important Topics

    • Marcel Berger

    Pages 659-691

16. The Technical Chapter

    • Marcel Berger

    Pages 693-721

17. Back Matter

    Pages 723-824

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Riemannian geometry has today become a vast and important subject. This new book of Marcel Berger sets out to introduce readers to most of the living topics of the field and convey them quickly to the main results known to date. These results are stated without detailed proofs but the main ideas involved are described and motivated. This enables the reader to obtain a sweeping panoramic view of almost the entirety of the field. However, since a Riemannian manifold is, even initially, a subtle object, appealing to highly non-natural concepts, the first three chapters devote themselves to introducing the various concepts and tools of Riemannian geometry in the most natural and motivating way, following in particular Gauss and Riemann.

• Laplace-Beltrami operator
• Ricci flow
• Riemannian geometry
• Riemannian manifold
• analysis on manifolds
• curvature
• differential geometry
• ergodic theory
• geodesic flow
• manifold

From the reviews:

"In this monumental work, Marcel Berger manages to survey large parts of present day Riemannian geometry. … the book offers a great opportunity to get a first impression of some part of Riemannian geometry, together with hints for further reading." (A.Cap, Monatshefte für Mathematik, Vol. 145 (4), 2005)

"Riemannian geometry has become a vast subject, influencing, famously, the development of general relativity and, more recently, the classification of 3-manifolds by hyperbolic structures … . Marcel Berger’s book is an overview of this enormous subject. … Virtually everything is illustrated with clear and useful diagrams … . This is the sort of book one could dip into or refer to over a period of years." (Peter Giblin, The Mathematical Gazette, March, 2005)

"Marcel Berger’s A Panoramic View of Riemannian Geometry is without doubt the most comprehensive, original and idiosyncratic treatise on differential geometry … . he manages to include the most up-to-date references on even the most classical topics that he presents, and he puts far greater emphasis on applications. … the book concludes with a massive and extremely useful bibliography of 1310 items." (Robert Osserman, SIAM Review, Vol. 47 (1), 2005)

"This book of one of the main contributors to Riemannian geometry has as a first goal to give an overview to most of the living topics of the subject. … A second goal is to show how many intuitive geometric questions lead to Riemannian geometry in a natural way. In my opinion it is this way of explaining … which makes the book highly recommendable to students as well as to experienced geometers. Furthermore, this book is clearly an encyclopedia in Riemannian geometry … ." (F.Manhart, Internationale Mathematische Nachrichten, Issue 197, 2004)

"This book is really a panorama. … the reading creates pleasure for the interested reader. … the book has intrinsic value for a student as well as for an experienced geometer. Additionally, it is really a compendium in Riemannian Geometry." (Jürgen Eichhorn, Mathematical Reviews, 2004 h)

"Riemannian geometry has today become a vast and important subject. This new book of Marcel Berger sets out to introduce readers to most of the living topics of the field and convey them quickly to the main results known to date. … enables the reader to obtain a sweeping panoramic view of almost the entirety of the field." (L'ENSEIGNEMENT MATHEMATIQUE, Vol. 49 (3-4), 2003)

• Institut des Hautes Études Scientifiques IHES, Bures-sur-Yvette, France

  Marcel Berger

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