Global Differential Geometry

Primary reference

1. Riemannian Manifolds: An Introduction to Curvature - John M. Lee.

Other useful references

1. Riemannian Geometry - M. P. do Carmo.

2. Riemannian Geometry: A modern Introduction - I. Chavel.

3. A comprehensive introduction to Differential Geometry - M. Spivak

Grading policy, etc.

• Homework will be assigned regularly, roughly once every two-three weeks.

• You have the option of working on a project and presenting your work, or a take home final. If you choose to work on a project, that work can be extended in the spring for Math 537B.

• Here is a list of potential projects. You can pick one of these topics or come talk to me if you have some ideas you would like to investigate.
  • Embedding negatively curved surfaces in R^3, Hilbert's theorem and Efimov's theorem
  • Convex integration, smooth and rough embeddings, Nash-Kuiper theorem, Gromov's theorem.
  • Chern-Gauss-Bonnet theorem
  • Finsler geometry, geodesics and curvature
  • Sub-Riemannian geometry, applications to control theory
  • Symplectic geometry and classical mechanics
  • Lorentzian geometry, Einstein field equations and the global structure of space-time
  • Discretizing manifolds and operators, Laplace-Beltrami operator and curvature.
  • Geometric flows and geometric variational problems, applications to image processing.
  • Sobolev spaces on manifolds; Harmonic maps and wave maps.

Homework

Homework 1

Homework 2

Homework 3

Homework 4