An Introduction to Manifolds

1. Front Matter

   Pages i-xviii

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2. A Brief Introduction

   • Loring W. Tu

   Pages 1-2

3. Euclidean Spaces

   • Loring W. Tu

   Pages 3-45

4. Manifolds

   • Loring W. Tu

   Pages 47-83

5. The Tangent Space

   • Loring W. Tu

   Pages 85-162

6. Lie Groups and Lie Algebras

   • Loring W. Tu

   Pages 163-188

7. Differential Forms

   • Loring W. Tu

   Pages 189-234

8. Integration

   • Loring W. Tu

   Pages 235-272

9. De Rham Theory

   • Loring W. Tu

   Pages 273-316

10. Back Matter

    Pages 317-410

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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, 'Introduction to Manifolds' is also an excellent foundation for Springer's GTM 82, 'Differential Forms in Algebraic Topology'.

• 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

From the reviews of the second edition:

“This book could be called a prequel to the book ‘Differential forms in algebraic topology’ by R. Bott and the author. Assuming only basic background in analysis and algebra, the book offers a rather gentle introduction to smooth manifolds and differential forms offering the necessary background to understand and compute deRham cohomology. … The text also contains many exercises … for the ambitious reader.” (A. Cap, Monatshefte für Mathematik, Vol. 161 (3), October, 2010)

• , Department of Mathematics, Tufts University, Medford, USA

  Loring W. Tu

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 Professor of Mathematics at Tufts University in Massachusetts. An algebraic geometer by training, he has done research at 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."

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