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Differentiable manifolds. A first course. (English) Zbl 0770.57001
Basler Lehrbücher. Boston, MA: Birkhäuser. xii, 395 p. (1993).
Table of contents: 1) Topological Manifolds, 2) Local Theory, 3) Global Theory, 4) Flows and Foliations, 5) Lie Groups, 6) Covectors and 1-Forms, 7) Multilinear Algebra, 8) Integration and Cohomology, 9) Forms and Foliations, 10) Riemannian Geometry; Appendices: A. Vector Fields on Spheres, B. Inverse Function Theorem, C. Ordinary Differential Equations, D. Sard’s Theorem, E. de Rham-Čech Theorem.
It is not an easy task to write now a new book on differentiable manifolds. The existing literature is abundant. However, I should say that the author has very well succeeded in writing an interesting, stimulating and pleasant reading book. Naturally it has nonvoid intersections with other classical texts. But the spirit of the book has a distinguished, personal flavour. There is an intelligent equilibrium between rigor and in formal, deep results (such as those refering to vector fields on spheres or the classification of compact connected surfaces) are presented without (or with sketchy) proofs but with large comments. I should also mention the brief yet illuminating exposition of Riemannian geometry, though, personally I felt the lack of the Gauss- Bonnet theorem.
The main idea of the book is to gradually persuade the student that local and global phenomena are complementary interrelated. Passing from nonlinear problems to linear ones through differentiation, the use of linear algebra, then integrating to recover the original nonlinear data is a process explained and emphasized throughout the text. Numerous examples and exercises make clear these ideas.
Although labeled as “first course” the book is intented to graduate students. The reader should have a good background in general topology and linear algebra. Besides, the aim of the author was not to provide an introduction in the domain but to “prepare the students for advanced topics courses and seminars in differential topology and geometry” (from the Preface). He surely reached this goal and much more.
The review cannot be exhaustive, but it can gain readers for the book. I hope it did so.

MSC:
57-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes
57R99 Differential topology
58A05 Differentiable manifolds, foundations
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