Mathematics G4402. Modern Geometry I, Fall 2012

Instructor: Chiu-Chu Melissa Liu
Office: Mathematics 623
Phone: 212-854-2499
E-mail: ccliu@math.columbia.edu, ccl2121@columbia.edu
Office Hours: Tuesday 4:30-5:30pm, Wednesday 3:30-4:30pm, or by appointments Melissa Liu's schedule in Fall 2012

Lectures: Monday & Wednesday, 8:40-9:55pm, Mathematics 407

Teaching Assistant: Tristan Clifford Collins
Office: Mathematics 408
Phone: 212-854-5881
E-mail: tcollins@math.columbia.edu, tcc2119@columbia.edu

Roundtables: Friday 5-6pm, Room 622.

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

References:

• [Bo] William M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, revised second edition
• [GHL] Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine, Riemannian Geometry , third edition

Syllabus:

• Differential manifolds ([dC] Chapter 0, [GHL] 1.A-1.H):
  smooth manifolds, smooth maps (submersions, immersions, embeddings), tangent spaces and tangent bundles, vector fields (flows, Frobenius theorem), Lie groups and homogeneous spaces, covering maps and fibrations, tensors, differential forms and the Stokes theorem
• Riemannian Geometry ([dC] Chapter 1-4, [GHL] 2.A, 2.B, 2.C, 3.A):
  Riemannian metrics, affine connections, Riemannian connections, geodesics, curvature

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

• Homework: There will be 10 assignments, due in class on Mondays.
  Due dates: (1)9/17 (2)9/24 (3)10/1 (4)10/15 (5)10/22 (6)11/7 (7)11/19 (8) 11/26 (9)12/3
  Note that Assignment 6 is due on a Wednesday.
  The total score of each assignment is 50; the late penalty is -4 points per day.
  Assignment 0: Richard Schoen's plenary talk at ICM 2010, "Riemannian manifolds of positive curvature" (due Mon.9/12)
  Assignment 1 (due 9/17), Assignment 2 (due 9/24), Assignment 3 (due 10/1), Assignment 4 (due 10/15),
  Assignment 5 (due 10/22), Assignment 6 (due Wed.11/7), Assignment 7 (due 11/19), Assignment 8 (due 11/26),
  Assignment 9 (due 12/3), Assignment 10
• Midterm: Wednesday, October 24, 2012, in class
• Final Exam: Wednesday, December 19, 2012, 9am-noon (scheduled by the university).