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...P jzp xpre cunyb angel x april pre cb chavel february char count this page intentionally left blank ii riemanniangeometry amodernintroduction second edition book provides an introduction to riemannian geometry the of curvedspaces foruseinagraduatecourse requiringonlyanunderstandingof differentiable manifolds covers introductory ideas followed by a selection more specialized topics also featured are notes and exercises for each chapter develop enrich reader s appreciationofthesubject thissecondeditionhasaclearertreatmentofmany from rst with new proofs some theorems on surfaces has been added main themes here effect curvature usual notions classical euclidean motivated cur vatureitself amongtheclassicaltopicsshowninanewsettingisisoperimetric inequalities interplay volume sets areas their bound aries in curved space completely created include rauch comparison theorem its consequences topology interaction microscopic behavior macroscopic structure isaac is professor mathematics at city col...