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**Introduction to differentiable manifolds.
(Introduction aux variétés différentielles.)**
*(French)*
Zbl 0872.53001

Collection Grenoble Sciences. Grenoble: Presses Universitaires de Grenoble. 299 p. (1996).

It isn’t an easy task to write another textbook on differentiable manifolds: several good texts are available (Warner’s, Postnikov’s, Conlon’s etc.). Still there is place for new, well-written items.

That is the case with the text under review. The author’s declared aim was to write a text that “tout en restant élémentaire, aborde les variétés le plus directement possible en s’intéressant principalement à leurs propriétés topologiques” (from the Introduction). He succeeded. The main achievement of the book is the clear, stylish, attractive presentation of the material, mostly elementary, but giving also some insight on more advanced topics in the “Commentaires” paragraph which ends each chapter. Some comments regarding applications in physics are welcome, too, e.g. the paragraph on Maxwell’s equations in Chapter V, Differential forms. It is worth noting several “non-standard” proofs for some classical results, such as the classification of one-manifolds or Whitney’s embedding theorem for compact manifolds.

The book contains all the topics it should contain: an introductory chapter on calculus in \({\mathbb{R}}^n\), manifolds (local and global theory), basic Lie groups, differential forms, theory of integration, ending with a thirty page chapter on cohomology (without sheaves) and degree theory. Very important, each chapter ends with a consistent section of exercises (146 in all), most of them with hints or solutions at the end of the book.

Summing up, a pleasant reading for the specialist, a very useful text for the student. Physicists and any others interested in differential geometry would also benefit. A good-to-have book in a library.

That is the case with the text under review. The author’s declared aim was to write a text that “tout en restant élémentaire, aborde les variétés le plus directement possible en s’intéressant principalement à leurs propriétés topologiques” (from the Introduction). He succeeded. The main achievement of the book is the clear, stylish, attractive presentation of the material, mostly elementary, but giving also some insight on more advanced topics in the “Commentaires” paragraph which ends each chapter. Some comments regarding applications in physics are welcome, too, e.g. the paragraph on Maxwell’s equations in Chapter V, Differential forms. It is worth noting several “non-standard” proofs for some classical results, such as the classification of one-manifolds or Whitney’s embedding theorem for compact manifolds.

The book contains all the topics it should contain: an introductory chapter on calculus in \({\mathbb{R}}^n\), manifolds (local and global theory), basic Lie groups, differential forms, theory of integration, ending with a thirty page chapter on cohomology (without sheaves) and degree theory. Very important, each chapter ends with a consistent section of exercises (146 in all), most of them with hints or solutions at the end of the book.

Summing up, a pleasant reading for the specialist, a very useful text for the student. Physicists and any others interested in differential geometry would also benefit. A good-to-have book in a library.

Reviewer: L.Ornea (Bucureşti)