In this second edition, the main additions are a section devoted to surfaces with constant negative curvature, and an introduction to conformal geometry. Also, we present a -soft-proof of the Paul Levy-Gromov isoperimetric inequal ity, kindly communicated by G. Besson. Several people helped us to find bugs in the. first edition. They are not responsible for the persisting ones! Among them, we particularly thank Pierre Arnoux and Stefano Marchiafava. We are also indebted to Marc Troyanov for valuable comments and sugges tions. INTRODUCTION This book is an outgrowth of graduate lectures given by two of us in Paris. We assume that the reader has already heard a little about differential manifolds. At some very precise points, we also use the basic vocabulary of representation theory, or some elementary notions about homotopy. Now and then, some remarks and comments use more elaborate theories. Such passages are inserted between *. In most textbooks about Riemannian geometry, the starting point is the local theory of embedded surfaces. Here we begin directly with the so-called "abstract" manifolds. To illustrate our point of view, a series of examples is developed each time a new definition or theorem occurs. Thus, the reader will meet a detailed recurrent study of spheres, tori, real and complex projective spaces, and compact Lie groups equipped with bi-invariant metrics. Notice that all these examples, although very common, are not so easy to realize (except the first) as Riemannian submanifolds of Euclidean spaces.
Rezensionen / Stimmen
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 therelations 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 für Mathematik, Vol. 147 (1), 2006)
Reihe
Auflage
Sprache
Verlagsort
Verlagsgruppe
Illustrationen
ISBN-13
978-3-642-97242-3 (9783642972423)
DOI
10.1007/978-3-642-97242-3
Schweitzer Klassifikation
I. Differential Manifolds.- A. From Submanifolds to Abstract Manifolds.- B. Tangent Bundle.- C. Vector Fields.- D. Baby Lie Groups.- E. Covering Maps and Fibrations.- F. Tensors.- A characterization for tensors.- G. Exterior Forms.- H. Appendix: Partitions of Unity.- II. Riemannian Metrics.- A. Existence Theorems and First Examples.- B. Covariant Derivative.- C. Geodesics.- Definitions.- III. Curvature.- A. The Curvature Tensor.- B. First and Second Variation of Arc-Length and Energy.- C. Jacobi Vector Fields.- E. The Behavior of Length and Energy in the Neighborhood of a Geodesic.- F. Manifolds with Constant Sectional Curvature.- G. Topology and Curvature.- H. Curvature and Volume.- I. Curvature and Growth of the Fundamental Group.- J. Curvature and Topology: An Account of Some Old and Recent Results.- K. Curvature Tensors and Representations of the Orthogonal Group.- L. Hyperbolic Geometry.- M. Conformai Geometry.- IV. Analysis on Manifolds and the Ricci Curvature.- A. Manifolds withBoundary.- B. Bishop's Inequality Revisited.- C. Differential Forms and Cohomology.- A second visit to the Bochner method.- D. Basic Spectral Geometry.- E. Some Examples of Spectra.- F. The Minimax Principle.- G. The Ricci Curvature and Eigenvalues Estimates.- H. Paul Levy's Isoperimetric Inequality.- V. Riemannian Submanifolds.- A. Curvature of Submanifolds.- B. Curvature and Convexity.- C. Minimal Surfaces.- Some Extra Problems.- Solutions of Exercises.- I.- II.- III.- IV.- V.
