Differential Geometry I (2016 Fall)

Room: 綜三631
Time: Wednesday 678 (14:30-15:30, 15:40-16:30, 16:40-17:20)
Text book (for the most part): L.Tu, An Introduction to Manifolds, UTX.

Course Description

The material that we hope to cover in the course includes (but not restricted to) the following:

1. Smooth manifolds
2. Tangent spaces and cotangent spaces
3. Vector Bundles
4. Differential forms
5. Integration on manifolds
6. De Rham cohomology, Hodge theory, exact sequences
7. Lie groups, group actions on manifolds

However, the exact material is not determined at this point. In principal, there will be Differential Geometry II in the Spring term, so whatever is not finished in the Fall, will be continued plus new material that will be disclosed later.

References

M. do Carmo, Differential forms and applications, UTX.
D. Barden and C. Thomas, An Introduction to Differential Manifolds.
S. Morita, Geometry of Differential forms.
S. Kobayashi, Fundations of Differential Geometry I & II.
Frank Warner, Foundations of Differentiable Manifolds and Lie Groups, GTM.
W.Boothby, An Introduction to Differential Manifolds and Riemannian Geometry.
V.Guillemin and A.Pollack, Differential Topology.
I.M.Singer and J.A.Thorpe, Lecture notes on Elementary Topology and Geometry, UTM.
R.Bott and L.Tu, Differential Forms in Algebraic Topology, GTM.
M.Spivak, A Comprehensive Introduction to Differential Geometry I.

Syllabus

9/14: Review topology, manifolds (examples including quotients)(Appendix A, section 5, section 7)
9/21: Quotients (continuing), maps on and between manifolds (section 7, 6)
9/28: National Holiday
10/5: Tangent vectors as derivations, tangent space (section 8, 9) (required: section 2)
10/12: Constant rank theorems, regular value theorem (Homework will be given today) (section 9, 11) (required: Appendix B)
10/19: Vector bundles, tangent bundle, partitions of unity (section 12)
10/26: Vector fields and their properties (Hand in your homework today) (section 13, 14)
11/2: Lie groups and Lie algebra, cotangent bundle, differential 1-forms, tensor
11/9: Midterm I (up to section 14)
11/16: University Holiday
11/23: Differential k-forms, Exterior derivative (section 17, 18) (required section 3, 4)
11/30: Operators on differential forms, Orientations, Integration on manifolds (section 19, 20, 21)
12/7: Integration on manifolds, Manifolds with boundary, Stokes Theorem (Homework will be given today) (section 22, 23)
12/14: Move to another day (12/30? 1/12?)
12/21: de Rham cohomology(Hand in your homework today)(section 24, 25)
12/28: Poincare Lemma, Homotopy operator (section 26, 27)(Poincare Lemma can be found in do Carmo or Morita's books)
12/30: Poincare Lemma, Homotopy operator (section 26, 27)(Poincare Lemma can be found in do Carmo or Morita's books)
1/4: Final Exam (up to section 26)
1/11: Poincare duality (Jost's Riemannian geometry and geometric analysis)

Evaluation

Midterm 50%, Final exam 50%.
Homework is not calculated in the final mark, it serves the purpose of practicing how to write a proper proof. Moreover, it might be considered if/when one is at the boundary, so I do recommend you hand in the homework.

Last Updated: Jan 11, 2017
URL: http://www.math.nthu.edu.tw/~nankuo/DG2016.html