© 2008

An Introduction to Manifolds

  • Assumes only one semester of algebra and two semesters of undergraduate analysis, so is accessible to undergraduates and graduate students

  • Modesty of scope allows the most essential topics to emerge clearly

  • Provides hints and solutions to about half of the exercises, so may be used in the classroom or for independent study

  • Includes an introduction to the necessary topics in point-set topology, so book is self-contained and suitable for a wide audience


Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Euclidean Spaces

  3. Manifolds

    1. Pages 47-55
    2. Pages 63-74
  4. Lie Groups and Lie Algebras

    1. Pages 77-89
    2. Pages 91-100
    3. Pages 101-104
    4. Pages 105-117
    5. Pages 119-126
    6. Pages 135-146
  5. Lie Groups and Lie Algebras

    1. Pages 149-160
    2. Pages 161-171
  6. Differential Forms

    1. Pages 175-179
    2. Pages 181-188
    3. Pages 189-198

About this book


Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory.

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; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems.

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. Requiring only minimal undergraduate prerequisites, An Introduction to Manifolds is also an excellent foundation for Springer GTM 82, Differential Forms in Algebraic Topology.



Algebraic topology De Rham cohomology Homotopy cohomology exterior derivative homology manifold

Authors and affiliations

  1. 1.Tufts UniversityMedfordUSA

About the authors

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).

Bibliographic information


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)