Mathematics GR6402. Modern Geometry I, Fall 2016

Meeting Time: MW 10:10-11:25am Meeting Location: Mathematics 520

Instructor: Chiu-Chu Melissa Liu
E-mail: ccliu@math.columbia.edu, ccl2121@columbia.edu
Office: Mathematics 623
Phone: 212-854-2499
Office Hours: Monday 4:15-5:15pm, Tuesday 5-6pm, Wednesday 3-4pm, Thursday 3-4pm, Friday 5-6pm, or by appointments Melissa Liu's schedule in Fall 2016

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 1-5, [GHL] 2.A, 2.B, 2.C, 3.A):
  Riemannian metrics, affine connections, Riemannian connections, geodesics, curvature, Jacobi fields

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

• Homework: There will be weekly assignments, due on the following dates.
  (1)Mon.9/19 (2)Mon.9/26 (3)Mon.10/3 (4)Mon.10/10 (5)Mon.10/17 (6)Mon.10/24
  (6)Wed.11/9 (7)Mon.11/14 (8)Mon.11/21 (9)Mon.11/28 (11)Mon.12/5 (12)Mon.12/12
  The total score of each assignment is 50; the late penalty is -2 points per day.
  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 10/24), Assignment 7 (due 11/9), Assignment 8 (due 11/14),
  Assignment 9 (due 11/21), Assignment 10 (due 11/28), Assignment 11 (due 12/5), Assignment 12 (due 12/12)
• Midterm: Wednesday, October 26, 2016, in class.
  The midterm will cover lectures before October 20, and Assignment 1-6.
• Final Exam: Monday, December 19, 2016, 9am-noon (scheduled by the University Registrar).
  The final exam will cover everything taught in this class, and all the assignments.