Many years have passed since the ?rst edition. However, the encouragements of various readers and friends have persuaded us to write this third edition. During these years, Riemannian Geometry has undergone many dramatic - velopments. Here is not the place to relate them. The reader can consult for instance the recent book [Br5]. of our "mentor" Marcel Berger. However, R- mannian Geometry is not only a fascinating ?eld in itself. It has proved to be a precious tool in other parts of mathematics. In this respect, we can quote the major breakthroughs in four-dimensional topology which occurred in the eighties and the nineties of the last century (see for instance [L2]). These have been followed, quite recently, by a possibly successful approach to the Poincar´ e conjecture. In another direction, Geometric Group Theory, a very active ?eld nowadays (cf. [Gr6]), borrows many ideas from Riemannian or metric geometry. Butletusstophoggingthelimelight.Thisisjustatextbook.Wehopethatour point of view of working intrinsically with manifolds as early as possible, and testingeverynewnotiononaseriesofrecurrentexamples(seetheintroduction to the ?rst edition for a detailed description), can be useful both to beginners and to mathematicians from other ?elds, wanting to acquire some feeling for the subject.
From the reviews of the third edition:
"This new edition maintains the clear written style of the original, including many illustrations ... examples and exercises (most with solutions)." (Joseph E. Borzellino, Mathematical Reviews, 2005)
"This book based on graduate course on Riemannian geometry ... covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features of the subject. Classical results ... are treated in detail. ... contains numerous exercises with full solutions and a series of detailed examples which are picked up repeatedly to illustrate each new definition or property introduced. For this third edition, some topics ... have been added and worked out in the same spirit." (L'ENSEIGNEMENT MATHEMATIQUE, Vol. 50, (3-4), 2004)
"This book is based on a graduate course on Riemannian geometry and analysis on manifolds that was held in Paris. ... Classical results on the relations between curvature and topology are treated in detail. The book is almost self-contained, assuming in general only basic calculus. It contains nontrivial exercises with full solutions at the end. Properties are always illustrated by many detailed examples." (EMS Newsletter, December 2005)
"The guiding line of this by now classic introduction to Riemannian geometry is an in-depth study of each newly introduced concept on the basis of a number of reoccurring well-chosen examples ... . The book continues to be an excellent choice for an introduction to the central ideas of Riemannian geometry." (M. Kunzinger, Monatshefte fur Mathematik, Vol. 147 (1), 2006)