Includes bibliographical references (pages 131-163) and indexes.
0. Riemannian geometry up to 1950 1. Comments on the main topics I, II, III, IV, V under consideration Part I. Curvature and topology Part II. The geometrical hierarchy of Riemann manifolds: Space forms Part III. The set of Riemannian structures on a given compact manifold: Is there a best metric? Part IV. The spectrum, the eigenfunctions Part V. Periodic geodesics, the geodesic flow TOP. Some other Riemannian geometric topics of interest