This textbook for secondyear graduate students is intended as an introduction to differential geometry with principal emphasis on Riemannian geometry. Chapter I explains basic definitions and gives the proofs of the important theorems of Whitney and Sard. Chapter II deals with vector fields and differential forms. Chapter III addresses integration of vector fields and \(p\)plane fields. Chapter IV develops the notion of connection on a Riemannian manifold considered as a means to define parallel transport on the manifold. The author also discusses related notions of torsion and curvature, and gives a working knowledge of the covariant derivative. Chapter V specializes on Riemannian manifolds by deducing global properties from local properties of curvature, the final goal being to determine the manifold completely. Chapter VI explores some problems in PDEs suggested by the geometry of manifolds. The author is well known for his significant contributions to the field of geometry and PDEsparticularly for his work on the Yamabe problemand for his expository accounts on the subject. The text contains many problems and solutions, permitting the reader to apply the theorems and to see concrete developments of the abstract theory. Readership Graduate students, research mathematicians, and mathematics educators interested in differential geometry. Reviews "More than half of the book is devoted to exercises, problems at different levels and solutions of exercises. The author's aim was to facilitate the teaching of differential geometry. The presentation is very successful, and I can strongly recommend the book to anybody willing to learn differential geometry, as well as to teachers of the subject."  European Mathematical Society Newsletter "The author is one of the best contemporary geometers and draws from his extended experience in selecting the topics and the various approaches ... It covers topics every working mathematician (or theoretical physicist) ought to know ... The style is very clear and concise, and the emphasis is not on the widest generality, but on the most often encountered situation. This makes it a much more approachable text than many other traditional sources ... an excellent textbook for a first course on basic differential geometry, very helpful to both the instructors and their students."  Mathematical Reviews Table of Contents  Background material
 Differentiable manifolds
 Tangent space
 Integration of vector fields and differential forms
 Linear connections
 Riemannian manifolds
 The Yamabe problemAn introduction to research
 Bibliography
 Subject index
 Notation
