Riemannian Geometry
A Modern Introduction

This corrected and clarified second edition, first published in 2006, includes a new chapter on the Riemannian geometry of surfaces.

Isaac Chavel (Author)

9780521853682, Cambridge University Press

Hardback, published 10 April 2006

488 pages, 161 exercises
22.9 x 15.2 x 2.7 cm, 0.82 kg

'… I think that it is the best reference on Riemannian geometry available, especially for someone interested in isoperimetric problems. … an insightful modern perspective on topics of current research interest.' SIAM Review

This book provides an introduction to Riemannian geometry, the geometry of curved spaces, for use in a graduate course. Requiring only an understanding of differentiable manifolds, the author covers the introductory ideas of Riemannian geometry followed by a selection of more specialized topics. Also featured are Notes and Exercises for each chapter, to develop and enrich the reader's appreciation of the subject. This second edition, first published in 2006, has a clearer treatment of many topics than the first edition, with new proofs of some theorems and a new chapter on the Riemannian geometry of surfaces. The main themes here are the effect of the curvature on the usual notions of classical Euclidean geometry, and the new notions and ideas motivated by curvature itself. Completely new themes created by curvature include the classical Rauch comparison theorem and its consequences in geometry and topology, and the interaction of microscopic behavior of the geometry with the macroscopic structure of the space.

1. Riemannian manifolds
2. Riemannian curvature
3. Riemannian volume
4. Riemannian coverings
5. Surfaces
6. Isoperimetric inequalities (constant curvature)
7. The kinetic density
8. Isoperimetric inequalities (variable curvature)
9. Comparison and finiteness theorems.

Subject Areas: Differential & Riemannian geometry [PBMP], Euclidean geometry [PBMH], Calculus & mathematical analysis [PBK]