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AuthorGallot, Sylvestre. author
TitleRiemannian Geometry [electronic resource] / by Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1990
Edition Second Edition
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Descript XIII, 286 p. online resource


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 inequalยญ 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 suggesยญ 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


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 with Boundary -- B. Bishopโ{128}{153}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 Levyโ{128}{153}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

Mathematics Differential geometry Manifolds (Mathematics) Complex manifolds Mathematics Differential Geometry Manifolds and Cell Complexes (incl. Diff.Topology)


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