Differential Geometry I (2024 Fall)

• Room: 綜三203
• Time: Friday 567
• Office Hour: By appointment
• Text book (for the most part): L.Tu, An Introduction to Manifolds, UTX.
• 助教: 黃祈叡 (Room 212) Office Hour: By appointment
• 請同學在暑假期間閱讀第一到第四章，以及附錄AB這兩章，複習大學已經學過且這學期會需要用到的內容
• 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。
• 下面的進度只是預估(以*標記預估進度，完成的會將*移除)。考試日期與範圍也是預估，可能會有改變，請同學到時候注意網頁的UPDATE

Course Description

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

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

• 9/6: Manifolds, quotients (section 5,7) (required: Appendix A)
• 9/13: Maps on and between manifolds, Tangent vectors as derivations, tangent space, differential of smooth maps (section 6, 8) (required: section 2)
• 9/20: Constant rank theorems, regular value theorem, submanifolds (section 9, 11,) (required: Appendix B)
• 9/27: Embedding, vector bundles, tangent bundle (section 12, 13) (Homework: read Appendix C)
• 10/4: Partition of unity, Vector fields and their properties (section 13, 14)
• 10/11: Lie groups and Lie algebra (section 15, 16)
• 10/18: Cotangent bundle, differential 1-forms, Tensor, Differential k-forms (section 17, 18) (required section 3, 4)
• 10/25: Exterior derivative, operators on differential forms (section 19, 20) (required section 3,4)
• 11/1: Midterm I (up to section 16)
• 11/8: Contraction, Lie derivative on differential forms (section 20)
• 11/15: Orientations (section 21)
• 11/22: Manifolds with boundary, Integration on manifolds (section 22, 23)
• 11/29: Stokes Theorem , de Rham cohomology (section 23, 24)
• 12/6: Exact sequences, Homotopy equivalence (section 25, 26, 27) (required: Appendix D)
• 12/13: Computation of de Rham cohomology (section 28, 29)
• 12/20: Final Exam All (以期中考後的範圍為主)
• Further study: Topics
• 12/30: 助教辦公室212室下午2-4點看考卷

Exercise

• section 5: #1
• section 6: #1
• section 7: #2,5,9
• section 8: #1,7
• section 9: #3,7,10
• section 11: #2,3
• section 12: #4
• section 13: #5
• section 14: #2,12,13
• section 15: #4,7,15
• section 16: #3,8,10
• section 17: #3,4
• section 18: #3,8,9
• section 19: #1,3,8,10,11
• section 20: #4,6,8,9
• section 21: #4,5,6,7
• section 22: #5,9,10,11
• section 23: #1,5
• section 24: #1
• section 25: #3,4
• section 26: #1,2
• section 27: #1,3
• section 28: #1,2,3,4
• section 29: #3

Evaluation

• Midterm 50%, Final Exam 50%