Lectures: Wednesday, 10:20 ~ 11:10, and Friday 10:20 ~ 12:10; at Astro-Math 101
Office hour: 14:00 ~ 15:00 every Wednesday; at Astro-Math 458
Course Assistant: 李冠輝
Problem session: Wednesday 11:20 ~ 12:10; at Astro-Math 101
Office hour: Thursday 15:30 ~ 17:00; at Astro-Math 441
Course prerequisite: linear algebra (with theory) and mathematical analysis
It will be helpful if you are familiar with general topology. (more precisely, ch.1 of Topology and Geometry by Glen Bredon mathscinet)
Grading scheme:
Homework 25%
You have two jokers: the lowest two grades will be discarded.
Midterm I 30%
Midterm II 30%
Final Exam 15%
Main Topics:
Theory of surfaces
Differentiable manifolds
Differential forms and vector bundles
Textbooks:
[MR] Sebastián Montiel and Antonio Ros, Curves and surfaces. Second Edition mathscinet (a)
[DFN2] Boris A. Dubrovin, Anatoliĭ T. Fomenko and Sergeĭ P. Novikov, Modern geometry---methods and applications. Part II. The geometry and topology of manifolds. mathscinet (b)
[T] Loring W. Tu, Differential geometry. Connections, curvature, and characteristic classes. mathscinet (c)
Other references:
[DFN1] Boris A. Dubrovin, Anatoliĭ T. Fomenko and Sergeĭ P. Novikov, Modern geometry---methods and applications. Part I. The geometry of surfaces. mathscinet
[DFN3] Boris A. Dubrovin, Anatoliĭ T. Fomenko and Sergeĭ P. Novikov, Modern geometry---methods and applications. Part III. Introduction to homology theory. mathscinet
[dC] Manfredo P. do Carmo, Differential geometry of curves and surfaces. mathscinet (a)
[C] Shiing-Shen Chern, Wei-Huan Chen and Kai-Shue Lam, Lectures on differential geometry. mathscinet (b)
[GP] Victor Guillemin and Alan Pollack, Differential topology. mathscinet (b)
[BT] Raoul Bott and Loring W. Tu, Differential forms in algebraic topology. mathscinet (c)
[M] Shigeyuki Morita, Geometry of differential forms. mathscinet (c)
Lecture summaries and references
(Week 1) plane curves and space curves. Reference: [MR, ch.1], and note
(Week 2) regular surface, tangent plane and smooth map, facts of compact regular surfaces. Reference: [MR, ch.2 and ch.4], and note
(Week 3) Gauss map, first and second fundamental forms, Gaussian curvature and mean curvature. Reference: [MR, ch.3 and ch.7], and note
(Week 4) Gauss Theorema Egregium, degree of a map to S^2. Reference: [MR, ch.7 and ch.8], and note and note
(Week 5) degree of a map to S^2 (continued), global Gauss--Bonnet theorem (pt.1). Reference: [MR, ch.8], and note
(Week 6) Poincaré--Hopf theorem, global Gauss--Bonnet theorem (pt.2). Reference: [MR, ch.8], and note
(Week 7) geodesic. Reference: [MR, ch.7], and note
Midterm I.
(Week 8) smooth manifold. Reference: [DFN2, §1~§5 of ch.1], and note
(Week 9) derivation and tangent bundle, smooth map: embedding and immersion. Reference: [GP, §2~§4 of ch.1], and note
(Week 10) Whitney embedding and baby Morse theory. Reference: [DFN2, §11 of ch.2], [GP, §7~§8 of ch.1], note and note
(Week 11) Lie bracket and Frobenius theorem. Reference: [C, §1-4], and note and note
(Week 12) introduction to Lie groups. Reference: [DFN2, §3 of ch.1], [T, §15] and note
(Week 13) example of root system: special unitary group. Reference: note
Midterm II.
(Week 14) exterior algebra and differential form. Reference: note
(Week 15) Stokes theorem and Poincaré lemma. Reference: [BT, §I.3 and §I.4], and note
(Week 16) Mayer--Vietoris sequence and argument. Reference: [BT, §I.2 and §I.5], and note