The book under review is divided into 8 chapters and at the end of each chapter, some exercises are given. The author gives a synthesis of geometric and analytic methods in the study of Riemannian manifolds. All the fundamental concepts are defined. The following questions are treated: Differentiable and Riemannian manifolds, vector bundles, vector fields and one-parameter groups of diffeomorphisms, tensor calculus, the Hodge theorem for de Rham cohomology, connections, curvature, parallel transport, metric connections, the Yang-Mills functional, Levi-Civita connection, submanifolds, geodesics, Jacobi fields, the Rauch comparison theorem, Morse theory and applications to the existence of closed geodesics, the theorem of Lyusternik and Fet on closed geodesics, the Palais-Smale condition and closed geodesics, symmetric spaces, Kähler manifolds, harmonic maps and the Bochner technique, higher regularity, harmonic maps into manifolds of nonpositive curvature.
The exposition is clear, well organized and generally easy to read. The book uses both invariant global calculus (intrinsic) and tensor notations. There are over 80 exercises in the book. The sections entitled “Perspectives” include historical remarks and references, and place the results of the book in a broader context without giving detailed proofs. Sobolev spaces and regularity theory for linear elliptic equations are discussed. The geometric analysis of this book “is to a large extent due to the work and the influence of Shing-Tung Yau”.