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
  7. Integration

    1. Pages 201-209
    2. Pages 211-220
  8. De Rham Theory

  9. Back Matter
    Pages 280-364

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

  • Loring W. Tu
    • 1
  1. 1.Tufts UniversityMedfordUSA

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