Geometry II, Spring 2019

Course Information

• Ceiba
• Lectures: Wednesday, 10:20 ~ 11:30, and Friday 10:20 ~ 11:30; at Astro-Math 101
• Office hours: 14:00 ~ 15:00 every Wednesday; at Astro-Math 458
• Grading scheme:
  • Homework 30%
    You have two jokers: the lowest two grades will be discarded.
  • Midterm 30%
  • Final 30%
  • in-class discussion 10%
• Course prerequisite:
  • (undergraduate) algebra and analysis
  • manifold, differential form
• Main Topics:
  • vector bundle: connection, curvature, characteristic class
  • Riemannian geometry: geodesic, Jacobi field, geometry of submanifold
  • to be decided later, maybe some elliptic PDE stuff
• Textbooks:
  • [M] Shigeyuki Morita, Geometry of differential forms. mathscinet
  • [P] Peter Petersen, Riemannian geometry. mathscinet
  • [CE] Jeff Cheeger and David Ebin, Comparison theorems in Riemannian Geometry. mathscinet

Lecture summaries and references

• (Week 1) vector bundle: definition and basic examples. Reference: [M, §5.1], and note
• (Week 2) vector bundle: more constructions. Reference: [M, §5.1], and note
• (Week 3) connection. Reference: [M, §5.2], and note
• (Week 4) curvature. Reference: [M, §5.3 and §5.4], and note
• (Week 5) characteristic class. Reference: [M, §5.4 and §5.5], and note
• (Week 6) Euler class and more. Reference: [M, §5.6 and §5.7], and note
  Midterm: 10:20 ~ 12:10, March 29.
• (Week 7) 清明連假
• (Week 8) Riemannian metric, Levi-Civita connection, Riemann curvature tensor. Reference: [P, §2.2 and §3.1], and note
• (Week 9) geodesics, first variational formula, Gauss lemma. Reference: [P, §5.4 and §5.5], and note
• (Week 10) Hopf--Rinow theorem, second variational formula. Reference: [P, §5.7 and §6.1], [CE, §1.3 and §1.6], and note
• (Week 11) 自主學習週
• (Week 12) Final Exam, file
• (Week 13) geometry of submanifold: Gauss, Codazzi, Ricci equations. the method of moving frame. Reference: [P, §3.2], note and note
• (Week 14) tensor calculus in classical notation, variation of the Hilbert functional. Reference: note
• (Week 15) constant Gaussian curvature on surfaces: analysis of Laplacian. Reference: note
• (Week 16) constant Gaussian curvature on surfaces: solving the nonlinear PDE. Reference: note
• (Week 17) introduction to Hodge theory. Reference: note

Homework

• Homework 01: due February 27.
• Homework 02: due March 6.
• Homework 03: due March 13.
• Homework 04: due March 20.
• Homework 05: due March 27.
• Homework 06: due April 17.
• Homework 07: due April 24.
• Homework 08: due May 22.
• Homework 09: due May 29.
• Homework 10: due June 12.