Differential Geometry I (2017 Fall)

Course Description

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

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, Foundations 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.

9/13: Manifolds, maps on and between manifolds (section 5, section 6)
9/20: Review Topology, quotients (section 7, Appendix A)
9/27: Tangent vectors as derivations, tangent space (section 8, 9) (required: section 2)
10/4: National Holiday
10/11: Constant rank theorems, regular value theorem (section 9, 11) (required: Appendix B)
10/18: Vector bundles, tangent bundle, Vector fields (section 12, 14)
10/25: Partition of unity, Vector fields and their properties, Lie groups (section 13, 14, 15)
11/1: Midterm I (up to section 11)
11/8: Lie algebra, cotangent bundle, differential 1-forms (section 16, 17)
11/15: University Holiday
11/22: Differential 1-forms, tensor, Differential k-forms, (section 17, 18) (required section 3)
11/29: Exterior derivative, operators on differential forms (section 19, 20) (required section 3,4)
12/6: Midterm II (up to section 18)
12/13: Lie derivative, Orientations (section 20, 21)
12/20: Orientations, Manifolds with boundary (section 21, 22)
12/27: Integration on manifolds Integration on manifolds, Stokes Theorem (section 23)
1/3*: de Rham cohomology (section 24, 25)
1/10*: Final Exam (up to section ?)

Last Updated: December 27, 2017
URL: http://www.math.nthu.edu.tw/~nankuo/DG2017.html