Mathematics G4402. Modern Geometry I, Fall 2014

Meeting Time: MW 10:10-11:25am Meeting Location: Mathematics 507

Instructor: Chiu-Chu Melissa Liu
Office: Mathematics 623
Phone: 212-854-2499
E-mail: ccliu@math.columbia.edu, ccl2121@columbia.edu
Office Hours: Tuesday & Thursday 5-6pm, or by appointments Melissa Liu's schedule in Fall 2014

Teaching Assistant: Zijun Zhou
Email: zzhou@math.columbia.edu, zz2224@columbia.edu
Office: Mathematics 408
Roundtables: Monday 5:40-6:40pm, Mathematics 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 and differential forms
• 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 12 assignments, due in class on the following dates.
  (1)Mon.9/15 (2)Mon.9/22 (3)Mon.9/29 (4)Mon.10/6 (5)Mon.10/13 (6)Mon.10/20
  (7)Wed.10/29 (8)Mon.11/10 (9)Mon.11/17 (10)Mon.11/24 (11) Wed.12/3 (12) Mon.12/8
  The total score of each assignment is 50; the late penalty is -3 points per day.
  Assignment 1 (due 9/15), Assignment 2 (due 9/22), Assignment 3 (due 9/29),
  Assignment 4 (due 10/6), Assignment 5 (due 10/13), Assignment 6 (due 10/20),
  Assignment 7 (due 10/29), Assignment 8 (due 11/10), Assignment 9 (due 11/17),
  Assignment 10 (due 11/24), Assignment 11 (due Wed.12/3), Assignment 12 (due Mon.12/8)
• Midterm: Wednesday, October 22, 2014, in class
• Final Exam: Monday, December 15, 2014, 9am-noon (scheduled by the University Registrar).