Traditional point of view: pinched manifolds 147 Almost flat pinching 148 Coarse point of view: compactness theorems of Gromov and Cheeger 149 K. CURVATURE AND REPRESENTATIONS OF THE ORTHOGONAL GROUP Decomposition of the space of curvature tensors 150 Conformally flat manifolds 153 The second Bianchi identity 154 CHAPITRE IV : ANALYSIS ON MANIFOLDS AND THE RICCI CURVATURE A. MANIFOLDS WITH BOUNDARY Definition 155 The Stokes theorem and integration by parts 156 B. BISHOP'S INEQUALITY REVISITED 159 Some commutations formulas Laplacian of the distance function 160 Another proof of Bishop's inequality 161 The Heintze-Karcher inequality 162 C. DIFFERENTIAL FORMS AND COHOMOLOGY The de Rham complex 164 Differential operators and their formal adjoints 165 The Hodge-de Rham theorem 167 A second visit to the Bochner method 168 D. BASIC SPECTRAL GEOMETRY 170 The Laplace operator and the wave equation Statement of the basic results on the spectrum 172 E. SOME EXAMPLES OF SPECTRA 172 Introduction The spectrum of flat tori 174 175 Spectrum of (sn, can) F. THE MINIMAX PRINCIPLE 177 The basic statements VIII G. THE RICCI CURVATURE AND EIGENVALUES ESTIMATES Introduction 181 Bishop's inequality and coarse estimates 181 Some consequences of Bishop's theorem 182 Lower bounds for the first eigenvalue 184 CHAPTER V : RIEMANNIAN SUBMANIFOLDS A. CURVATURE OF SUBMANIFOLDS Introduction 185 Second fundamental form 185 Curvature of hypersurfaces 187 Application to explicit computations of curvature 189 B. CURVATURE AND CONVEXITY 192 The Hadamard theorem C.
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)
