Mathematics G4403. Modern Geometry II, Spring 2016

Meeting Time: MW 10:10-11:25am Meeting Location: Mathematics 307

Instructor: Chiu-Chu Melissa Liu
Office: Mathematics 623
Phone: 212-854-2499
E-mail: ccliu@math.columbia.edu, ccl2121@columbia.edu
Office Hours: Monday 4:15-5:15pm, Tuesday 5-6pm, Wednesday 3-4pm, Thursday 1:30-2:30pm, Friday 5-6pm, or by appointments

Textbook: [dC] Manfredo Perdigao do Carmo, Riemannian Geometry

References:

• [GHL] Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine, Riemannian Geometry, third edition.
• [CE] Jeff Cheeger and David G. Ebin, Comparison Theorems in Riemannian Geometry.
• [KN] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of Differential Geometry I & II.
• [M] Shigeyuki Morita, The Geometry of Differential Forms .
• [PT] Thomas Parker, Clifford Taubes, "On Witten's Proof of the Positive Energy Theorem," Commu. Math. Phys. {\bf 84} (1982), no.2, 223-238.
• [W] Edward Witten, ``A new proof of the positive energy theorem," Comm. Math. Phys. {\bf 80} (1981), no. 3, 381-402

Tentative Syllabus:

• Riemannian Geometry (continued).
  isometric immersions (second fundamental forms), complete manifolds, Hopf-Rinow and Cartan-Hadamard theorems, manifolds of constant sectional curvature, first and second variations of energy, Bonnet-Myers theorem and Synge-Wienstein theorem
  References: [dC] Chapter 6-9, [GHL] Section 3.B, 3.D-3.M, 5.A, [CE] Chapter 1
• Principal Bundles.
  principal bundles and associate bundles, connections and curvatures on a principal bundle, induced connections and curvatures on associated vector bundles, parallel transport and holonomy, characteristic classes
  References: [KN] Chapter I Section 5, Chapter II, Chapter XII, [M] Chapter 5, 6
• Witten's Proof of the Positive Energy Theorem. References: [PT], [W]

Grading: Homework 40%, Midterm 20%, Final 40%

• Homework: There will be weekly assignments, due in class on Mondays.
  The total score of each assignment is 50. The lowest homework score will be dropped.
  Assignment 14 (due 2/1), Assignment 15 (due 2/8), Assignment 16 (due 2/17), Assignment 17 (due 2/29),
  Assignment 18 (due 3/7), Assignment 19 (due 3/21), Assignment 20 (due 3/30), Assignment 21 (due 4/11),
  Assignment 22 (due 4/18), Assignment 23 (due 4/25), Assignment 24 (due 5/2)
• Midterm: Wednesday, March 9, in class.