A Panoramic View of Riemannian Geometry

In his latest book A Panoramic View of Riemannian Geometry, Marcel Berger does a remarkable job giving an in-depth survey ranging over almost the full spectrum of Riemannian geometry, furnishing the reader with some of the most exciting and elegant topics from classical to modern, and from local to global Riemannian geometry.

The reader will find that, unlike many other books on Riemannian geometry, the style of this 15-chapter 824-page manuscript is quite unique. The author does not follow the traditional definition-theorem-proof approach; instead, he only motivates and presents, without detailed proofs, the best possible results in many areas known to date, thereby providing interested readers with a valuable source and efficient means for learning about the latest advances. Additionally, throughout the book, open problems are introduced and discussed as soon as they can be stated.

The organization of the material in this book also distinguishes itself from many of its peers — as Berger says, "in our division into chapters, necessarily arbitrary, we did not follow any logical or historical order. We have tried to follow certain naturalness and simplicity." The contents of the book are divided into the following chapters:

1. Euclidean Geometry — "Old and New Euclidean Geometry and Analysis"
2. Transition — "The Need for a More General Framework"
3. Surfaces from Gauss to Today
4. Riemann's Blueprints — "Riemann's Blueprints for Architecture in Myriad Dimensions"
5. A One Page Panorama
6. Metric Geometry and Curvature — "Riemannian Manifolds as Metric Spaces and the Geometric Meaning of Sectional and Ricci Curvature"
7. Volume and Inequalities on Volumes of Cycles
8. Transition: The Next Two Chapters
9. Spectrum of the Laplacian — "Riemannian Manifolds as Quantum Mechanical Worlds: The Spectrum and Eigenfunctions of the Laplacian"
10. Geodesic Dynamics — "Riemannian Manifolds as Dynamical Systems: the Geodesic Flow and Periodic Geodesics"
11. Best Metric — "What is the Best Riemannian Metric on a Compact Manifold?"
12. From Curvature to Topology
13. Holonomy Groups and Kähler Manifolds
14. Some Other Important Topics
15. The Technical Chapter

The first chapter on Euclidean geometry should be a delight to read for a broad audience. Its abundant collection of fascinating results, including some not-so-well-known theorems from classical curve and surface theory in Euclidean space, is even accessible to undergraduates. It may serve well as supplementary reading material for an introductory differential geometry course. The subsequent chapters are perhaps more suitable for those who have some advanced background knowledge in differential geometry. The last three chapters are relatively brief, and yet they provide a suitable overview of the main structures and results from these very current topics.

I believe many researchers with interests in Riemannian geometry as well as those who appreciate or want to learn what's current in Riemannian geometry may find this book beneficial. However, this book should not be, as cautioned by the author, used as a handbook or primer of Riemannian geometry; rather, with careful selection according to individual's taste, it can be a great reference which may enlarge the breadth of one's knowledge and enhance one's research.

Tan Zhang (tan.zhang@murraystate.edu) is assistant professor of mathematics at Murray State University in Murray, KY. His research interests are differential geometry and algebraic topology.