Differential Geometry II

Differential geometry is the study of Riemannian manifolds and their local and global properties.

In this course, we will cover some topics in differential geometry, possibly including:

• Calculus of variations and Morse theory on the space of paths
• The Cartan-Hadamard theorem and the geometry of nonpositively curved manifolds
• The geometry of Lie groups and symmetric spaces
• Comparison geometry

Basic information

• Instructor: Robert Young (ryoung@cims.nyu.edu)
• Office: CIWW 601
• Office hours: Mondays, 1:30--2:30 or by appointment
• Lectures: CIWW 201, Wednesdays 1:25--3:15
• Sources:
  • Milnor, Morse theory
  • Lee, Riemannian Manifolds: An Introduction to Curvature
  • Bridson and Haefliger, Metric Spaces of Non-Positive Curvature
  • Cheeger and Ebin, Comparison Theorems in Riemannian Geometry

Notes

• Week 1-2
• A different proof that flat manifolds are locally euclidean can be found in Riemannian Manifolds, page 119-121.
• Week 3-7
• Week 8-10
• Week 11-13

Problem Sets

• Problem Set 1 (due February 17)
• Problem Set 2 (due March 2)
• Problem Set 3 (due April 13)
• Problem Set 4 (due May 4)