Differential Geometry I (2013 Fall)

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Instructor: Rung-Tzung Huang
Room: M-430
Time: W234
Office hours: W56

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Textbook: Introduction to smooth manifolds, Second Edition, GTM, by John M. Lee, Link.
Reference: An introduction to manifolds, Second Edition, UTX, by Loring W. Tu, Link.

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Course Description:

The materials that will be covered in the course:

1. Smooth manifolds
2. Tangent and cotangent spaces
3. Vector bundles
4. Submanifolds
5. Sard's theorem(without proof), Whitney embedding theorem(without proof)
6. Lie groups
7. Vector fields, integral curves, flows, distributions, Frobenius Theorem(without proof)
8. Tensors and differential forms
9. Orientations and integrations on manifolds
10. de Rham cohomology

Lecture schedule:

1. 9/18:   Review of topology, definition of topological manifolds, examples of topological manifolds.
               Suggested reading: Chapter 1, Introduction to topological manifolds, Seond Edition, UTX, by John Lee, Link.
               Homework 1: Problem 1-1(Lee), Due date: 9/25.

2. 9/25:   Review of quotient topology, projective space, compatibility of coordinate charts.
               Homework 2: Exercise 7.11(Tu) and Problem 7.9 (Tu), Due date: 10/2.

3. 10/2:   Smooth functions and smooth maps, partition of unity.
               Homework 3: Problem 6.1(Tu), Due date: 10/9.

4. 10/9:   Partition of unity (continued), tangent vectors, the differential of a smooth map.
               Reading: Read the part "computations in coordinates" (page 60-65) in Lee's book.
               Homework 4: Problem 13.3(a) (Tu) and Exercise 3.17 (Lee), Due date: 10/16.

5. 10/16: Submersion, immersion, embedding, submanifolds.
               Homework 5: Problem 5-4(Lee), Due date: 10/23. (cf. Example 5.4 Tu, P.49.)

6. 10/23: Submanifolds (continued), Sard's theorem(without proof), Whitney theorem(without proof).
               Homework 6: Problem 5-7(Lee), Due date: 10/30.  

7. 10/30: Tangent bundle, vector fields, integral curves.
               Homework 7: Problem 8-10(Lee), 8-16(b)(Lee),  Due date:  11/6.

8. 11/6:  Integral curves(continued), flows, Lie derivative.
               Homework 8: Problem 9-3(Lee), Due date: 11/13.

9. 11/13: Lie derivative(continued), distributions, integral manifolds, Frobenius Theorem(without proof).
               Homework 9: Problem 19-5(Lee), Due date: 11/27.

10. 11/20: No class, University holiday.

11. 11/27: Multilinear algebra, symmetric and alternating tensors, the algebra of alternating tensors.
                Homework 10: Problem 12-5(Lee), 12-7(Lee), Due date: 12/4.

12. 12/4: Vector bundles, smooth sections, local frames.
                Homework 11: Problem 17.1(Tu), Due date: 12/11.

13. 12/11: Differential forms, definitions of orientations.
                 Homework 12: Problem 19.1, 19.3(Tu), Due date: 12/18.

14. 12/18: Orientations of manifolds, integration of differential forms, Stoke's theorem.
                 Homework 13: Problem 16-2(Lee), Due date: 12/25.

15. 12/25: de Rham cohomology, homotopy invariance of de Rham cohomology.
                 Homework 14: Problem 17-1(Lee), Due date: 1/8.

16. 1/1/2014: No class, National holiday.

17. 1/8: Lie groups,  Lie algebras associated to Lie groups, exponential maps,  Linear Lie  groups.

18. 1/15: Final oral presentations. 


Evaluation: 60% Homeworks, Attendances + 40% Final presentation