Geometry, Fall 2018

Course Information

• Ceiba
• 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:
  1. Theory of surfaces
  2. Differentiable manifolds
  3. 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

Homework

• Homework 00: do not have to submit.
• Homework 01: due Wednesday, September 19.
• Homework 02: due Wednesday, September 26.
• Homework 03: due Wednesday, October 3.
• Homework 04: due Friday, October 12.
• Homework 05: due Wednesday, October 17.
• Homework 06: due Wednesday, October 24.
• Homework 07: due Wednesday, November 7.
• Homework 08: due Wednesday, November 14.
• Homework 09: due Wednesday, November 21.
• Homework 10: due Wednesday, November 28.
• Homework 11: due Wednesday, December 5.
• Homework 12: due Wednesday, December 19.
• Homework 13: due Wednesday, December 26.