Mathematics G4402. Modern Geometry I, Fall 2015

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 2015

Teaching Assistant: Mitchell Faulk
Email: faulk@math.columbia.edu, mf2954@columbia.edu
Office: Mathematics 206
Phone: (212) 854-5886
Roundatable: TBA

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, 5):
  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)Thu.9/17 (2)Wed.9/23 (3)Wed.9/30 (4)Wed.10/7 (5)Wed.10/14 (6)Wed.10/21
  (7)Wed.11/4 (8)Thu.11/12 (9)Wed.11/18 (10)Wed.11/25 (11) Wed.12/2 (12) Wed.12/9
  The total score of each assignment is 50; the late penalty is -2 points per day.
  Assignment 1 (due 9/17), Assignment 2 (due 9/23), Assignment 3 (due 9/30), Assignment 4 (due 10/7),
  Assignment 5 (due 10/14), Assignment 6 (due 10/21), Assignment 7 (due 11/4), Assignment 8 (due 11/12),
  Assignment 9 (due 11/18), Assignment 10 (due 11/25), Assignment 11 (due 12/2), Assignment 12 (due 12/9),
  Assignment 13 (You are not required to submit solutions to Assignment 13.)
• Midterm: Monday, October 26, 2015, in class.
  The Midterm will cover the lectures before October 20, and Assignment 1-6.
• Final Exam: Monday, December 21, 2015, 9am-noon (scheduled by the University Registrar).
  The final exam will cover everything taught in this class, and all the assignments.