# An Introduction to Manifolds

• Loring W. Tu
Book

Part of the Universitext book series (UTX)

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
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Pages 163-188
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Pages 189-234
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Pages 235-272
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Pages 273-316
10. Back Matter
Pages 317-410

### 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