A Panoramic View of Riemannian Geometry

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.

Table of Contents:
1 Euclidean Geometry.- 2 Transition.- 3 Surfaces from Gauß to Today.- 4 Riemann’s Blueprints.- 5 A One Page Panorama.- 6 Metric Geometry and Curvature.- 7 Volumes and Inequalities on Volumes of Cycles.- 8 Transition: The Next Two Chapters.- 9 Spectrum of the Laplacian.- 10 Geodesic Dynamics.- 11 Best Metrics.- 12 From Curvature to Topology.- 13 Holonomy Groups and Kähler Manifolds.- 14 Some Other Important Topics.- 15 The Technical Chapter.- References.- Acknowledgements.- List of Notation.- List of Authors.

Review :
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 experiencedgeometer. 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)