Differential Geometry II (2014 Spring)

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Instructor: Rung-Tzung Huang
Room: M-219
Time: W234
Office hours: W56

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Textbook: Riemannian manifolds - an introduction to curvature, GTM, by John M. Lee, Link1 Link2

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Course Description:

The materials that will be covered in the course:

1. Introduction
2. Review of tensors, manifolds, and vector bundles
3. Definitions and examples of Riemannian metrics
4. Connections
5. Riemannian geodesics
6. Geodesics and distance
7. Curvature
8. Riemannian submanifolds
9. The Gauss-Bonnet theorem
10. Jacobi fields
11. Curvature and topology

Lecture schedule:

1. 2/19: Review of tensors and manifolds
             Reading: Chapter 1, What is curvature?

2. 2/26: Review of vector bundles, tensor bundles and tensor fields, definitions of Riemannian metrics
              HW1: Exercise 3.3-3.5,  Due date: 3/12

3. 3/5:   Examples of Riemannian metrics, elementary constructions associated with Riemannian metrics, gradident operator,
             Riemannian volume element, divergence and Laplace-Beltrami operators, the model space of Riemannian geometry
              HW2: Problem 3-3, 3-4, 3-5, Due date: 3/19

4. 3/12: Hyperbolic space, connections
              HW3: Problem 4-3, Due date: 3/19

5. 3/19: Existence of connections, covariant derivatives of tensor fields, vector fields along curves
              HW4: Problem 4-2, 4-5, Due date: 3/26

6. 3/26: Geodesics, parallel translation

7. 4/2:  The Riemannian (Levi-Civita) connection
              HW5: Exercise 5.2, 5.3, Problem 5-1, Due date: 4/9

8. 4/9:  The geodesic flow, the exponential map
              HW6: Problem 5-9, Due date: 4/23

9. 4/16: Oral presentations on homework problems I

10. 4/23: Normal neighborhoods and normal coordinates
              HW7: Excercise 5.6, Due date: 4/30

11. 4/30: Lengths and distances on Riemannian manifolds
              HW8: Problem 5-2, Due date: 5/7

12. 5/7:  Admissible families, minimizing cures are geodesics
              HW9: Problem 5-10, Due date: 5/14
 
13. 5/14: Geodesics are locally minimizing
              HW10: Problem 6-3, Due date: 5/21

14. 5/21: Completeness (Hopf-Rinow theorem)
              HW11: Problem 6-1,6-5, Due date: 5/28

15. 5/28: Curvature tensor, symmetries of the curvature tensor, Ricci curvature, scalar curvature, sectional curvature
              HW12: Problem 7-1,7-2, Due date: 6/4

16. 6/4: The model spaces of Riemannian spaces (homgeneous, isotropic, geodesics)

17. 6/11: Gauss-Bonnet theorem

18. 6/18: Oral presentations on homework problems II

Evaluation: Homeworks(including oral presentations on homework problems)+Attendances 100%