2006/04/10 by Isaac Chavel · 1 citation
Mathematics · #Absolute geometry #Computer science #Convex geometry #Curvature #Curvature of Riemannian manifolds #Differential geometry #Euclidean geometry #Euclidean space #Foundations of geometry #Geometry #Geometry and topology #History and Theory of Mathematics #Mathematical proof #Mathematics #Non-Euclidean geometry #Ordered geometry #Projective geometry #Pure mathematics #Regular polygon #Riemannian geometry #Scalar curvature #Sectional curvature #Space (punctuation) #Synthetic geometry
paper · doi:10.1017/cbo9780511616822
openalex publication_date 2006/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/22
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.