Geometry II, Spring 2019

Course Information

Ceiba (link retired)
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

(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