Differential Geometry I (2022 Fall)

• Room: 綜三201
• Time: Monday 456 (從第二周起改成12:30-3:20)
• Text book (for the most part): L.Tu, An Introduction to Manifolds, UTX.
• 助教: 蔡宗霖
• 請同學在暑假期間閱讀第一到第四章，複習大學已經學過的歐氏空間的情形
• 由於疫情發展未知，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，很可能會有改變，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, Hodge theory, 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

• 9/12: Review Topology, Manifolds, quotients (Appendix A, section 5, section 7)
• 9/19: Quotients, maps on and between manifolds (section 7, section 6)
• 9/26: Tangent vectors as derivations, tangent space (section 8, 9) (required: section 2)
• 10/3: Constant rank theorems, regular value theorem (section 9, 11) (required: Appendix B)
• 10/10: National Holiday
• 10/17: Vector bundles, tangent bundle, Vector fields, Partition of unity (section 12, 13, 14)
• 10/24: Vector fields and their properties, Lie groups and Lie algebra (section 14, 15)
• 10/31: Lie algebra, Cotangent bundle, differential 1-forms (section 16, 17)
• 11/7: Midterm I (up to section 14)
• 11/14: Differential 1-forms, tensor, Differential k-forms (section 17, 18) (required section 3)
• 11/21: Exterior derivative, operators on differential forms (section 19, 20) (required section 3,4)
• 11/28: Contraction, Lie derivative on differential forms (section 20)
• 12/5: Lie derivative on vector fields (section 20)
• 12/12: Midterm II (up to section 20)
• 12/19: Orientations, Manifolds with boundary (section 21, 22)
• 12/26: Integration on manifolds Integration on manifolds, Stokes Theorem (section 23)
• 1/2: National Holiday Stokes Theorem (Video)
• 1/9*: Final Exam (up to section 23)

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

Evaluation

• Midterm I 30%, Midterm II 30%, Final Exam 40%.