Differential Geometry I (2026 Fall)

Room: 綜三631
Time: Tuesday 678
Office Hour: By appointment
Text book (for the most part): L.Tu, An Introduction to Manifolds, UTX.
助教: 洪呈毅 習題課: W9 綜三203
請同學在暑假期間先讀第1到第4章及附錄A與B。1到4章是在歐氏空間上的情形，只要修過高微與線代即可了解，附錄A與B是複習高微學過的點集拓樸和反函數定理。有先複習大學已經學過且這學期會需要用到的內容，這樣在學習上會比較從容。
期中考及期末考除了公假、病假、喪假外，其餘事假均不接受。
公假要有假單，喪假請提供證明，病假需為不可抗拒之理由如住院等必須在考試的時候去醫院，並需在事後提供考試當時就診證明。
考試日期與範圍是預估，可能會有改變，請同學注意網頁的UPDATE

Course Description

This is an introductory course on Differential manifold. We wish to cover some fundamental and commonly used knowledge in varies fields such as Riemannian geometry, Symplectic geometry, etc. so that the students may have enough background for more advanced geometry courses. The material that will be covered in the course includes (but not restricted to) the following:

1. Smooth manifolds, maps between manifolds
2. Tangent spaces and cotangent spaces
3. Vector Bundles
4. Lie groups and Lie algebras
5. Differential forms
6. De Rham cohomology, Differential complex

Integration and Stokes's Theorem may be taught in the second semester, together with Hodge theory etc.

References

M. do Carmo, Differential forms and applications, UTX. (easy to read)
D. Barden and C. Thomas, An Introduction to Differential Manifolds. (easy to read)
S. Morita, Geometry of Differential forms. (easy to read)
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.

Syllabus

9/8: 出國開會(9/16 補課75分鐘+9/30補課75分鐘)
9/9: 習題課第一週不上習題課
9/15: Manifolds, quotients (section 5,7) (required: Appendix A)
9/16: 習題課(補課75分鐘) Maps on and between manifolds (section 6) (required: section 2)
9/22: (如果沒有好就需要暫停一次)Tangent vectors as derivations, tangent space, differential of smooth maps (section 8) (required: section 2)
9/23: 習題課助教上課
9/29: Constant rank theorems, regular value theorem, submanifolds (section 9) (required: Appendix B)
9/30: 習題課(補課75分鐘)Embedding, partition of unity(section 11, 13) (Homework: read Appendix C)
10/6: Constant rank theorems, regular value theorem, submanifolds (section 9)
10/7: 習題課助教上課
10/13: 出國開會，跟助教調課(11/11補課75分鐘+11/25補課75分鐘)，助教上一堂
10/14: 習題課助教上課
10/20: Vector bundles, tangent bundle, Vector fields and their properties (section 12, 14)
10/21: 習題課助教上課
10/27: Cotangent bundle, differential 1-forms, Tensor, Differential k-forms (section 17, 18) (required section 3, 4)
10/28: 習題課助教上課
11/3: Exterior derivative, operators on differential forms (section 19, 20) (required section 3,4)
11/4: 習題課助教上課
11/10: Midterm I
11/11: 習題課(補課75分鐘)
11/17:
11/18: 習題課助教上課
11/24:
11/25: 習題課(補課75分鐘)
12/1:
12/2: 習題課助教上課
12/8:
12/9: 習題課助教上課
12/15:
12/16: 習題課助教上課
12/22: Final Exam

Exercise

section 5:
section 6:
section 7:
section 8:
section 9:
section 11:
section 12:
section 13:
section 14:

Evaluation

Midterm 50%, Final Exam 50%

Last Updated: September 12, 2026
URL: http://www.math.nthu.edu.tw/~nankuo/DG2026F.html