Differential Geometry Connections, Curvature, and Characteristic Classes

Differential Geometry Connections, Curvature, and Characteristic Classes / by Loring W. Tu.

This text presents a graduate-level introduction to differential geometry for mathematics and physics students. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bun...

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Bibliographic Details
Main Author: Tu, Loring W. (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Cham : Springer International Publishing : Imprint: Springer, 2017.
Edition:1st ed. 2017.
Series:Graduate Texts in Mathematics, 275
Springer eBook Collection.
Subjects:
Differential geometry.
Algebraic geometry.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • Preface
  • Chapter 1. Curvature and Vector Fields
  • 1. Riemannian Manifolds
  • 2. Curves
  • 3. Surfaces in Space
  • 4. Directional Derivative in Euclidean Space
  • 5. The Shape Operator
  • 6. Affine Connections
  • 7. Vector Bundles
  • 8. Gauss's Theorema Egregium
  • 9. Generalizations to Hypersurfaces in Rn+1
  • Chapter 2. Curvature and Differential Forms
  • 10. Connections on a Vector Bundle
  • 11. Connection, Curvature, and Torsion Forms
  • 12. The Theorema Egregium Using Forms
  • Chapter 3. Geodesics
  • 13. More on Affine Connections
  • 14. Geodesics
  • 15. Exponential Maps
  • 16. Distance and Volume
  • 17. The Gauss-Bonnet Theorem
  • Chapter 4. Tools from Algebra and Topology
  • 18. The Tensor Product and the Dual Module
  • 19. The Exterior Power
  • 20. Operations on Vector Bundles
  • 21. Vector-Valued Forms
  • Chapter 5. Vector Bundles and Characteristic Classes
  • 22. Connections and Curvature Again
  • 23. Characteristic Classes
  • 24. Pontrjagin Classes
  • 25. The Euler Class and Chern Classes
  • 26. Some Applications of Characteristic Classes
  • Chapter 6. Principal Bundles and Characteristic Classes
  • 27. Principal Bundles
  • 28. Connections on a Principal Bundle
  • 29. Horizontal Distributions on a Frame Bundle
  • 30. Curvature on a Principal Bundle
  • 31. Covariant Derivative on a Principal Bundle
  • 32. Character Classes of Principal Bundles
  • A. Manifolds
  • B. Invariant Polynomials
  • Hints and Solutions to Selected End-of-Section Problems
  • List of Notations
  • References
  • Index.

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