An Introduction to Manifolds

- A Brief Introduction

Part I. The Euclidean Space
- Smooth functions on R(n)
- Tangent vectors in R(n) as derivations
- Alternating k-linear functions
- Differential forms on R(n)

Part II. Manifolds
- Manifolds
- Smooth maps on a manifold
- Quotient

Part III. The tangent space
- The tangent space
- Submanifolds
- Categories and functors
- The image of a smooth map
- The tangent bundle
- Bump functions and partitions of unity
- Vector fields

Part IV. Lie groups and Lie algebras
- Lie groups
- Lie algebras

Part V. Differential forms
- Differential 1-forms
- Differential k-forms
- The exterior derivative

Part VI. Integration
- Orientations
- Manifolds with boundary
- Integration on a manifold

Part VII. 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

- Appendix A. Point-set topology
- Appendix B. Inverse function theorem of R(n) and related results
- Appendix C. Existence of a partition of unity in general
- Appendix D. Solutions to selected exercises
- Bibliography
- Index

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).

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)