Introduction to smooth manifolds.
2nd revised ed.

*(English)*Zbl 1258.53002
Graduate Texts in Mathematics 218. New York, NY: Springer (ISBN 978-1-4419-9981-8/hbk; 978-1-4419-9982-5/ebook). xvi, 708 p. (2013).

This book with 22 chapters and four appendices is an introductory graduate-level textbook on smooth manifold theory (smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector field, flows, foliations, Lie derivatives, Lie groups, Lie algebras, quotient manifolds, symplectic manifolds and more). The appendices contain fundamental facts from basic topology, fundamental groups, covering spaces, a review of linear algebra, a review of calculus, a review of differential equations. The book also contains a large number of exercises and problems for the student to work out. At the end of each chapter, there is a collection of longer and harder questions (problems), very interesting for a full understanding of the theory.

The second edition of this book has been “extensively revised and the topics have been substantially rearranged” to improve the clarity of the text. The author introduces two important analytic tools: the rank theorem and the fundamental theorem on flows and use them throughout the book. A few new topics have been added: Sard’s theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, the study of first-order partial differential equations, a brief treatement of degree theory for smooth maps between compact manifolds, an introduction to contact structures. He also consolidated the introductory treatements of Lie groups, Riemannian metrics and a new chapter about symplectic manifolds. Manifolds with boundary are now treated systematically throughout the book.

All in all, the exposition is clearly and carefully written, contains a lot of pictures and intuitive explanations. It could be used by beginning graduate students who want to undertake a deeper study of specialized/new applications of smooth manifold theory.

The second edition of this book has been “extensively revised and the topics have been substantially rearranged” to improve the clarity of the text. The author introduces two important analytic tools: the rank theorem and the fundamental theorem on flows and use them throughout the book. A few new topics have been added: Sard’s theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, the study of first-order partial differential equations, a brief treatement of degree theory for smooth maps between compact manifolds, an introduction to contact structures. He also consolidated the introductory treatements of Lie groups, Riemannian metrics and a new chapter about symplectic manifolds. Manifolds with boundary are now treated systematically throughout the book.

All in all, the exposition is clearly and carefully written, contains a lot of pictures and intuitive explanations. It could be used by beginning graduate students who want to undertake a deeper study of specialized/new applications of smooth manifold theory.

Reviewer: Corina Mohorianu (Iaşi)

##### MSC:

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

53Cxx | Global differential geometry |

57Rxx | Differential topology |

58Axx | General theory of differentiable manifolds |