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An Introduction to Manifolds


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Table of Contents

Preface to the Second Edition.- Preface to the First Edition.-Chapter 1. Eudlidean Spaces. 1. Smooth Functions on a Euclidean Space.- 2. Tangent Vectors in R(N) as Derivativations.- 3. The Exterior Algebra of Multicovectors.- 4. Differential Forms on R(N).- Chapter 2. Manifolds.- 5. Manifolds.- 6. Smooth Maps on a Manifold.- 7. Quotients.- Chapter 3. The Tangent Space.- 8. The Tangent Space.- 9. Submanifolds.- 10. Categories and Functors.- 11. The Rank of a Smooth Map.- 12. The Tangent Bundle.- 13. Bump Functions and Partitions of Unity.- 14. Vector Fields.-Chapter 4. Lie Groups and Lie Algebras.- 15. Lie Groups.- 16. Lie Algebras.- Chapter 5. Differential Forms.- 17. Differential 1-Forms.- 18. Differential k-Forms.- 19. The Exterior Derivative.- 20. The Lie Derivative and Interior Multiplication.- Chapter 6. Integration.- 21. Orientations.- 22. Manifolds with Boundary.- 23. Integration on Manifolds.- Chapter 7. De Rham Theory.- 24. De Rham Cohomology.- 25. The Long Exact Sequence in Cohomology.- 26. The Mayer -Vietoris Sequence.- 27. Homotopy Invariance.- 28. Computation of de Rham Cohomology.- 29. Proof of Homotopy Invariance.-Appendices.- A. Point-Set Topology.- B. The Inverse Function Theorem on R(N) and Related Results.- C. Existence of a Partition of Unity in General.- D. Linear Algebra.- E. Quaternions and the Symplectic Group.- Solutions to Selected Exercises.- Hints and Solutions to Selected End-of-Section Problems.- List of Symbols.- References.- Index.

About the Author

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."


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 fur Mathematik, Vol. 161 (3), October, 2010)

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