Ebook: Riemannian Geometry

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.

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This book, based on a graduate course on Riemannian geometry and analysis onmanifolds, given in Paris, covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features on the subject. Classical results on the relations between curvature and topology are treated in detail. The bookis quite self-contained, assuming of the reader only a knowledge of differential calculus in Euclidean space. It contains numerous exercises with full solutions and a series of detailed examples which are picked up again repeatedly to illustrate each new definition or property introduced. This book addresses both the graduate student wanting to learn Riemannian geometry, and also the professional mathematician from a neighbouring field who needs information about ideas and techniques which are now pervading many parts of mathematics.

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This book, based on a graduate course on Riemannian geometry and analysis onmanifolds, given in Paris, covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features on the subject. Classical results on the relations between curvature and topology are treated in detail. The bookis quite self-contained, assuming of the reader only a knowledge of differential calculus in Euclidean space. It contains numerous exercises with full solutions and a series of detailed examples which are picked up again repeatedly to illustrate each new definition or property introduced. This book addresses both the graduate student wanting to learn Riemannian geometry, and also the professional mathematician from a neighbouring field who needs information about ideas and techniques which are now pervading many parts of mathematics.
Content:
Front Matter....Pages I-XIII
Differential Manifolds....Pages 1-50
Riemannian Metrics....Pages 51-105
Curvature....Pages 106-179
Analysis on Manifolds and the Ricci Curvature....Pages 180-215
Riemannian Submanifolds....Pages 216-231
Back Matter....Pages 232-286

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This book, based on a graduate course on Riemannian geometry and analysis onmanifolds, given in Paris, covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features on the subject. Classical results on the relations between curvature and topology are treated in detail. The bookis quite self-contained, assuming of the reader only a knowledge of differential calculus in Euclidean space. It contains numerous exercises with full solutions and a series of detailed examples which are picked up again repeatedly to illustrate each new definition or property introduced. This book addresses both the graduate student wanting to learn Riemannian geometry, and also the professional mathematician from a neighbouring field who needs information about ideas and techniques which are now pervading many parts of mathematics.
Content:
Front Matter....Pages I-XIII
Differential Manifolds....Pages 1-50
Riemannian Metrics....Pages 51-105
Curvature....Pages 106-179
Analysis on Manifolds and the Ricci Curvature....Pages 180-215
Riemannian Submanifolds....Pages 216-231
Back Matter....Pages 232-286
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