Riemannian Geometry: A Modern Introduction

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Cambridge University Press, Apr 10, 2006 - Mathematics
This book provides an introduction to Riemannian geometry, the geometry of curved spaces, for use in a graduate course. Requiring only an understanding of differentiable manifolds, the author covers the introductory ideas of Riemannian geometry followed by a selection of more specialized topics. Also featured are Notes and Exercises for each chapter, to develop and enrich the reader's appreciation of the subject. This second edition, first published in 2006, has a clearer treatment of many topics than the first edition, with new proofs of some theorems and a new chapter on the Riemannian geometry of surfaces. The main themes here are the effect of the curvature on the usual notions of classical Euclidean geometry, and the new notions and ideas motivated by curvature itself. Completely new themes created by curvature include the classical Rauch comparison theorem and its consequences in geometry and topology, and the interaction of microscopic behavior of the geometry with the macroscopic structure of the space.

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Page 10 - R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature, Comm. Pure Appl. Math.
Page 6 - Ueber die isoperimetrische Eigenschaft des Kreises auf der Kugeloberfläche und in der Ebene.
Page 18 - Existence of smooth embedded surfaces of prescribed genus that minimize parametric even elliptic functionals on 3-manifolds. J.
Page 16 - Randol, B. (1974). Small eigenvalues of the Laplace operator on compact Riemann surfaces. Bull. Am. Math. Soc. 80, 996-1000. Randol, B.

About the author (2006)

Isaac Chavel is Professor of Mathematics at The City College of the City University of New York. He received his Ph.D. in Mathematics from Yeshiva University under the direction of Professor Harry E. Rauch. He has published in international journals in the areas of differential geometry and partial differential equations, especially the Laplace and heat operators on Riemannian manifolds. His other books include Eigenvalues in Riemannian Geometry (1984) and Isoperimetric Inequalities: Differential Geometric and Analytic Perspectives (Cambridge University Press, 2001). He has been teaching at The City College of the City University of New York since 1970, and has been a member of the doctoral program of the City University of New York since 1976. He is a member of the American Mathematical Society.

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