Mathematics G4402. Modern Geometry I, Fall 2011

Instructor: Chiu-Chu Melissa Liu
E-mail: ccliu@math.columbia.edu, ccl2121@columbia.edu
Office: 623 Mathematics Building
Phone: (212) 854-2499
Office Hours: Monday 4-5:30pm, Wednesday 4-5pm, or by appointments: Melissa's schedule in Fall 2011

Lectures: Monday & Wednesday, 2:40-3:55pm, 307 Mathematics Building

Teaching Assistant: Ye-Kai Wang
E-mail: yw2293@math.columbia.edu, yw2293@columbia.edu
Office: 408 Mathematics Building
Phone: (212) 854-5881

Roundtables: Tuesday 11am-12noon, 622 Mathematics Building

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 1-6, [GHL] 2.A, 2.B, 2.C, 3.A, 5):
  Riemannian metrics, affine connections, Riemannian connections, geodesics, curvature, Jacobi fields, isometric immersions (second fundamental forms)

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

• Homework: There will be 11 assignments, due in class on Mondays. Due dates:
  (1) Mon.9/19 (2) Mon.9/26 (3) Mon.10/3 (4) Mon.10/10 (5) Mon.10/17
  (6) Mon.10/31 (7) Mon.11/14 (8) Mon.11/21 (9) Mon. 11/28 (10) Mon.12/5 (11) Mon.12/12
  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 9/19), Assignment 2 (due 9/26), Assignment 3 (due 10/3), Assignment 4 (due 10/10),
  Assignment 5 (due 10/17), Assignment 6 (due 10/31), Assignment 7 (due 11/14), Assignment 8 (due 11/21),
  Assignment 9 (due 11/28), Assignment 10 (due 12/5), Assignment 11 (due 12/12)
• Midterm: Wednesday, October 19, in class
• Final Exam: Wednesday, December 21, 1:10-4pm (scheduled by the university).