- Faculty, Mathematics
- Riemannian Geometry and Analysis, Laplacian and the Heat Operator on Riemannian Manifolds, Geometric Inequalities
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.
Published January 2006
Cambridge University Press, 2006
This introduction treats the classical isoperimetric inequality in Euclidean space and contrasting rough inequalities in noncompact Riemannian manifolds. In Euclidean space the emphasis is on a most general form of the inequality sufficiently precise to characterize the case of equality, and in Riemannian manifolds the emphasis is on those qualitiative features of the inequality that provide insight into the coarse geometry at infinity of Riemannian manifolds. The treatment in Euclidean space features a number of proofs of the classical inequality in increasing generality, providing in the process a transition from the methods of classical differential geometry to those of modern geometric measure theory; and the treatment in Riemannian manifolds features discretization techniques, and applications to upper bounds of large time heat diffusion in Riemannian manifolds. The result is an introduction to the rich tapestry of ideas and techniques of isoperimetric inequalities, a subject that has its beginnings in classical antiquity and which continues to inspire fresh ideas in geometry and analysis to this very day--and beyond!
Published July 2001
Cambridge University Press, 2001