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
• Comparison geometry
• The Cartan-Hadamard theorem and the geometry of nonpositively curved manifolds
• The geometry of Lie groups and symmetric spaces

Basic information

• Instructor: Robert Young (ryoung@cims.nyu.edu)
• Office: WWH 601
• Office hours: WWH 601, Wednesdays, 3:15--4:30
• Lectures: WWH 517, Wednesdays, 1:25--3:15
• Sources:
  • Milnor, Morse theory
  • Lee, Introduction to Riemannian Manifolds (Springer)
  • Cheeger and Ebin, Comparison Theorems in Riemannian Geometry
  • Bridson and Haefliger, Metric spaces of non-positive curvature

Notes

• March 11: Minimizing geodesics and the Morse index theorem
• May 6: Lower curvature bounds and Toponogov's Theorem

Problem sets

• Problem Set 1 (due February 19)
• Problem Set 2 (due February 26)
• Problem Set 3 (due March 11)
• Problem Set 4 (due March 25)
• Problem Set 5 (due May 13)