Differentiable Manifolds

ISBN-13: 9780817647667 / Angielski / Miękka / 2008 / 418 str.

Lawrence Conlon

Opis

The basics of differentiable manifolds, global calculus, differential geometry, and related topics constitute a core of information essential for the first or second year graduate student preparing for advanced courses and seminars in differential topology and geometry. Differentiable Manifolds is a text designed to cover this material in a careful and sufficiently detailed manner, presupposing only a good foundation in general topology, calculus, and modern algebra. This second edition contains a significant amount of new material, which, in addition to classroom use, will make it a useful reference text. Topics that can be omitted safely in a first course are clearly marked, making this edition easier to use for such a course, as well as for private study by non-specialists.

Recenzje

"This is a carefully written and wide-ranging textbook suitable mainly for graduate courses, although some advanced undergraduate courses may benefit from the early chapters. The subject matter is differential topology and geometry, that is, the study of curves, surfaces and manifolds where the assumption of differentiability adds the tools of differentiable and integral calculus to those of topology. Within this area, the book is unusually comprehensive.... The style is clear and precise, and this makes the book a good reference text. There are many good exercises."

-The Mathematical Gazette (Review of the Second Edition)

"This is a well-organized and nicely written text containing the basic topics needed for further work in differential geometry and global analysis.... This book is very suitable for students wishing to learn the subject, and interested teachers can find well-chosen and nicely presented materials for their courses."

-EMS Newsletter (Review of the Second Edition)

"This second edition contains a significant amount of new material, which, in addition to classroom use, will make it a useful reference text. Topics that can be omitted safely in a first course are clearly marked, making this edition easier to use for such a course, as well as for private study by non-specialists wishing to survey the field."

-L'enseignement Mathématique (Review of the Second Edition)

"The most outstanding difference between this book and other textbooks on differentiable manifolds is the emphasis on a very personal selection of topics in differential and algebraic topology. Overall, this edition contains more examples, exercises, and figures throughout the chapters."

-Mathematical Reviews (Review of the Second Edition)

"The author has very well succeeded in writing an interesting, stimulating and pleasant reading book [with] an intelligent equilibrium between rigor and informal."

-Zentralblatt Math (Review of the First Edition)

"While the first edition of this book is a 'first course' of differentiable manifolds, this edition contains a significant amount of new or expanded material . . . The book is well written, presupposing only a good foundation in general topology, calculus and modern algebra."

-Zentralblatt Math (Review of the Second Edition)

"The book under review is a new, enlarged and somewhat revised edition of the author's successful textbook Differentiable Manifolds (A first course) published also by Birkhäuser in 1993. Mathematicians already familiar with the earlier edition have spoken very favourably about the contents and the lucidity of the exposition. . . . In summary, this is an excellent and important book, carefully written and well produced. It will be a valuable aid to graduate and PhD students, lecturers, and-as a reference work-to research mathematicians."

-Publicationes Mathematicae (Review of the Second Edition)

"This book, revised and significantly expanded in comparison to the first edition, is devoted to foundations of differentiable manifolds, global calculus and differential geometry. It may serve as a basis for a two-semester graduate course for students of mathematics and as a reference book for graduate students of theoretical physics. . . . The choice of topics certainly gives the reader a good basis for further self study. . . . The book contains many interesting examples and exercises. The presentation is systematic and smooth and it is well balanced with respect to local versus global and between the coordinate free formulation and the corresponding expressions in local coordinates."

-Reports on Mathematical Physics (Review of the Second Edition)

"The author added a significant amount of new material, such that the book can become a reference book but it can still be used as a first course on differentiable manifolds. ...The book is useful for undergraduate and graduate students as well as for several researchers. The presentation is smooth, the choice of topics is optimal and the book can be profitably used for self teaching."

-Analele Stintifice ale Universitatii (Review of the Second Edition)

"This textbook, probably the best introduction to differential geometry to be published since Eisenhart's, greatly benefits from the author's knowledge of what to avoid, something that a beginner is likely to miss. The presentation is smooth, the choice of topics optimal, and the book can be profitably used for self teaching."

-The Bulletin of Mathematical Books (Review of the First Edition)

"A thorough, modern, and lucid treatment of the differential topology, geometry, and global analysis needed to begin advanced study of research in these areas. ...Conlon's book serves very well as a professional reference, providing an appropriate level of detail throughout. Recommended for advanced graduate students and above."

-Choice (Review of the First Edition)

"Probably the most outstanding novelty in such a book is the appropriate selection of topics from differential geometry, differential topology and algebraic topology."

-Mathematical Reviews (Review of the First Edition)

Spis treści

Preface to the Second Edition.-Topological Manifolds.-The Local Theory of Smooth Functions.-The Global Theory of Smooth Functions.-Flows and Foliations.-Lie Groups and Lie Algebras.-Covectors and 1--Forms.-Multilinear Algebra and Tensors.-Integration of Forms and de Rham Cohomology.-Forms and Foliations.-Riemannian Geometry.-Principal Bundles.-Appendix A. Construction of the Universal Covering.-Appendix B. Inverse Function Theorem.-Appendix C. Ordinary Differential Equations.-Appendix D. The de Rham Cohomology Theorem.-Bibliography.-Index.

Blurb

The basics of differentiable manifolds, global calculus, differential geometry, and related topics constitute a core of information essential for the first or second year graduate student preparing for advanced courses and seminars in differential topology and geometry. Differentiable Manifolds is a text designed to cover this material in a careful and sufficiently detailed manner, presupposing only a good foundation in general topology, calculus, and modern algebra. This second edition contains a significant amount of new material, which, in addition to classroom use, will make it a useful reference text. Topics that can be omitted safely in a first course are clearly marked, making this edition easier to use for such a course, as well as for private study by non-specialists wishing to survey the field.

The themes of linearization, (re) integration, and global versus local calculus are emphasized throughout. Additional features include a treatment of the elements of multivariable calculus, formulated to adapt readily to the global context, an exploration of bundle theory, and a further (optional) development of Lie theory than is customary in textbooks at this level. New to the second edition is a detailed treatment of covering spaces and the fundamental group.

Students, teachers and professionals in mathematics and mathematical physics should find this a most stimulating and useful text.

"This is a carefully written and wide-ranging textbook suitable mainly for graduate courses, although some advanced undergraduate courses may benefit from the early chapters. The subject matter is differential topology and geometry, that is, the study of curves, surfaces and manifolds where the assumption of differentiability adds the tools of differentiable and integral calculus to those of topology. Within this area, the book is unusually comprehensive…. The style is clear and precise, and this makes the book a good reference text. There are many good exercises."—The Mathematical Gazette (Review of the Second Edition)

"This textbook, probably the best introduction to differential geometry to be published since Eisenhart's, greatly benefits from the author's knowledge of what to avoid, something that a beginner is likely to miss. The presentation is smooth, the choice of topics optimal, and the book can be profitably used for self teaching."—The Bulletin of Mathematical Books (Review of the First Edition)

"The author has very well succeeded in writing an interesting, stimulating and pleasant reading book [with] an intelligent equilibrium between rigor and informal."—Zentralblatt Math (Review of the First Edition)