In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages, while other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems.
Rezensionen / Stimmen
From the reviews: "An introduction to the formalism of differential and integral calculus on smooth manifolds. ... Many prospective readers of Bott and Tu will welcome this volume. ... Summing Up: Recommended. Lower-division undergraduates." (D. V. Feldman, CHOICE, Vol. 45 (10), June, 2008) "An Introduction to Manifolds is split up into eight parts, well organized, well written, and, as Tu claims, readable. ... This excellent and accessible book also comes equipped with plenty of examples and exercises, whence it will serve well as both a classroom text and a source for self-study. Indeed, I propose to use it myself, given that I am one of the non-experts ... ." (Michael Berg, MathDL, April, 2008) "A book which ... covers all the essential topics in differentiable manifolds theory, and sufficiently elementary so that it can be read and understood with only minimal prerequisites-all this in less than 360 pages. The book is divided into seven parts, plus four appendices. ... The added value of the book lies mainly in the simplicity, the clearness and the concision of the exposition. ... is certainly one of the most readable introductions to differential geometry." (Ahmad El Soufi, Mathematical Reviews, Issue 2008 k) "The textbook under review is very well-written and self contained. ... It extends the calculus of curves and surfaces to higher dimensions. The higher dimensional analogues of smooth curves and surfaces are called manifolds. ... This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study." (Ion Mihai, Zentralblatt MATH, Vol. 1144, 2008)
Reihe
Auflage
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Graduate
Produkt-Hinweis
Illustrationen
104 s/w Abbildungen
1, black & white illustrations
Maße
Höhe: 23.5 cm
Breite: 15.5 cm
Dicke: 20 mm
Gewicht
ISBN-13
978-0-387-48098-5 (9780387480985)
DOI
10.1007/978-0-387-48101-2
Schweitzer Klassifikation
Loring W. Tu was born in Taipei, Taiwan, and grew up in Taiwan, Canada, and the United States. He attended McGill University and Princeton University as an undergraduate, and obtained his Ph.D. from Harvard University under the supervision of Phillip A. Griffiths. He has taught at the University of Michigan, Ann Arbor, and at Johns Hopkins University, and is currently on the faculty at Tufts University in Massachusetts.
An algebraic geometer by training, he has done research in the interface of algebraic geometry, topology, and differential geometry, including Hodge theory, degeneracy loci, moduli spaces of vector bundles, and equivariant cohomology. He is the coauthor with Raoul Bott of Differential Forms in Algebraic Topology (Springer Graduate Texts in Mathematics 82).
Euclidean Spaces.- Smooth Functions on a Euclidean Space.- Tangent Vectors in Rn as Derivations.- Alternating k-Linear Functions.- Differential Forms on Rn.- Manifolds.- Manifolds.- Smooth Maps on a Manifold.- Quotients.- Lie Groups and Lie Algebras.- The Tangent Space.- Submanifolds.- Categories and Functors.- The Rank of a Smooth Map.- The Tangent Bundle.- Bump Functions and Partitions of Unity.- Vector Fields.- Lie Groups and Lie Algebras.- Lie Groups.- Lie Algebras.- Differential Forms.- Differential 1-Forms.- Differential k-Forms.- The Exterior Derivative.- Integration.- Orientations.- Manifolds with Boundary.- Integration on a Manifold.- De Rham Theory.- De Rham Cohomology.- The Long Exact Sequence in Cohomology.- The Mayer-Vietoris Sequence.- Homotopy Invariance.- Computation of de Rham Cohomology.- Proof of Homotopy Invariance.
