An Introduction to Manifolds

  • Loring W. Tu

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Loring W. Tu
    Pages 1-2
  3. Loring W. Tu
    Pages 3-45
  4. Loring W. Tu
    Pages 47-83
  5. Loring W. Tu
    Pages 85-162
  6. Loring W. Tu
    Pages 163-188
  7. Loring W. Tu
    Pages 189-234
  8. Loring W. Tu
    Pages 235-272
  9. Loring W. Tu
    Pages 273-316
  10. Back Matter
    Pages 317-410

About this book

Introduction

Manifolds, the higher-dimensional analogues 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 second edition contains fifty pages of new material. Many passages have been rewritten, proofs simplified, and new examples and exercises added. This work may be used as a textbook for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. The requisite point-set topology is included in an appendix of twenty-five pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. Requiring only minimal undergraduate prerequisites, "An Introduction to Manifolds" is also an excellent foundation for the author's publication with Raoul Bott, "Differential Forms in Algebraic Topology."

Keywords

De Rham Theory Euclidean spaces Lie algebras Lie groups algebraic geometry degeneracy loci differential forms differential geometry geometric topology geometry of manifolds manifolds tangent space

Authors and affiliations

  • Loring W. Tu
    • 1
  1. 1., Department of MathematicsTufts UniversityMedfordUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4419-7400-6
  • Copyright Information Springer Science+Business Media, LLC 2011
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4419-7399-3
  • Online ISBN 978-1-4419-7400-6
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • About this book