This book is the second edition of the author's five-volume work, the first edition of which has been reviewed in detail [Vol. I, Spivak, Waltham, Mass., 1970; MR0267467; Vol. II, MR0271845; Vol. III, Publish or Perish, Boston, Mass., 1975; MR0372756; Vol. IV, V; MR0394452; MR0394453]. In the new edition, the basic plan of the book, as well as its contents, remains unchanged. However, in Volumes I and II, changes are made which improve the character of the exposition. First of all, all noticed errors are corrected. Some explanations are added to the first volume which are related to the exposition of the material in the subsequent volumes. In some cases, subsidiary material from the later volumes is transferred to the first volume. Some notations are changed in the first two volumes to correspond with the final publication of Volumes III, IV and V. Some propositions of a formal nature are moved from the main text to an appendix (for example, a theorem on the equivalence of tangent bundles from Chapter 1.3). The more difficult part of Riemann's memoir of 1861, concerning the curvature tensor, is moved from Chapter 4C of Volume II to the appendix to Chapter 6 of that volume. The appendix, entitled ``The Finsler metric'', is incorporated into the end of Chapter 4 of the second volume. The second part of the first appendix to Chapter 7 of Volume II is separated into a special second appendix. The layout of Volumes I and II is completely changed. They are now laid out like Volumes III, IV and V. All of the references on the pages of the later volumes are changed to correspond to the new pagination of the first two volumes. Some inaccuracies that were discovered in Volumes III-V are indicated on pages 447–449 of the third volume. In the bibliography, contained in Volume V, some books published after 1975 are added. Contrary to the assertion of the author, who calls his work ``notes'' on differential geometry, these five volumes completely deserve to be called ``books''. The main value of this book is the wide range of its study of differential geometry and the depth of its study of the field, as well as its detailed exposition of the fundamentals of this science. This book is distinguished from other books of its kind by its historical approach to the subject. The connection is made between the classical works of Euler, Gauss, Riemann and other geometers and the newest investigations in differential geometry. In regards to methodology, a particularly useful comparison of classical and contemporary methods for the exposition of specific problems of differential geometry is frequently applied. The material is further elucidated by the large number of original drawings that are in the text of the book. All of this, together with a huge quantity of examples scattered throughout the text and problems that are suggested for the reader to solve, make this book extremely useful to everyone who studies or teaches differential geometry.
Reviewer: Akivis, M. A. [form MathSciNet]