Differential Geometry: Connections, Curvature, and Characteristic Classes

by Loring W. Tu

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.

368 pages, Paperback

Published June 7, 2017