A panoramic view of Riemannian geometry. (English) Zbl 1038.53002

Berger, Marcel

Berlin: Springer (ISBN 3-540-65317-1/hbk). xxiii, 824 p. (2003).

This new book of Marcel Berger sets out to introduce readers to most of the living topics of the Riemannian geometry and convey them quickly to the main results known to date. For this reason, the author follows the only possible path: to present the results without proofs. He has two goals: first, to introduce the various concepts and tools of Riemannian geometry in the most natural way; or further, to demonstrate that one is practically forced to deal with abstract Riemannian manifolds in a host of intuitive geometrical questions. This explains why a long first chapter deals with problems in the Euclidean plane. Second, once equipped with the concept of Riemannian manifold, the author presents a panorama of current day Riemannian geometry. A panorama is never a full 360 degrees, but it is large enough to show the reader a substantial part of today’s Riemannian geometry. Open problems are presented as soon as they can be stated. This encourages the reader to appreciate the difficulty and the current state of each problem. The present book will bring pleasure and be of help to professional Riemannian geometers as well as those who want to enter into the realm of Riemannian geometry, which is an amazingly beautiful, active and natural field of research today.
From the Contents: Old and New Euclidean Geometry and Analysis. The Need for a More General Framework. Surfaces from Gauss to Today. Riemann’s Blueprints for Architecture in Myriad Dimensions. Riemannian Manifolds as Metric Spaces and the Geometric Meaning of Sectional and Ricci Curvature. Volumes and Inequalities on Volumes of Cycles. Riemannian Manifolds as Quantum Mechanical Worlds: The Spectrum and Eigenfunctions of the Laplacian. Riemannian Manifolds as Dynamical Systems: The Geodesic Flow and Periodic Geodesics. What is the Best Riemannian Metric on a Compact Manifold? From Curvature to Topology. Holonomy Groups and Kähler Manifolds. Noncompact Manifolds. Bundles over Riemannian Manifolds. Spinors. Harmonic Maps Between Riemannian Manifolds. Low Dimensional Riemannian Geometry. Orbifolds. The Technical Chapter.

Reviewer: Serguey M. Pokas (Odessa)

MSC:

53-02	Research exposition (monographs, survey articles) pertaining to differential geometry
53C20	Global Riemannian geometry, including pinching
53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions

Keywords:

Euclidean geometry; Riemannian geometry; Riemannian manifolds; metric spaces; plane curves; surfaces; inner geometry; Riemannian curvature tensor; sectional and Ricci curvature; spectrum and eigenfunctions of the Laplacian; dynamical systems; geodesic flow; periodic geodesics; compact manifold; holonomy groups; Kähler manifolds; bundles over Riemannian manifolds; spinors; harmonic maps; orbifolds