Math 268 Topics in Differential Geometry

Spring 1997

The basic material

The fundamentals of exterior differential systems will be developed, including integral manifolds, the Cartan-Kahler theory, and prolongation. Applications will include a study of the geometry of G-structures, the geometry of PDE (especially conservation laws) and variational problems, harmonic maps and morphisms, isometric embedding, and other classical problems from differential geometry as time permits.

Prerequisites

I will be assuming that the students are familiar with the material from Bill Allard's Math 267 from this fall. The outline for that course is as follows:

Calculus on manifolds
1. Manifolds
2. Jets and quantities
3. Vector fields and flows
4. Differential forms and exterior differentiation
5. Densities
6. Lie differentiation
Some applications
1. The general first order partial differential equation
2. Frobenius' theorem
3. Darboux' theorem
4. The one dimensional calculus of variations and a brief excursion into symplectic geometry
Introduction to Riemannian geometry
1. Connections on principal bundles
2. The bundle of orthonormal frames and the Riemannian curvature tensor
3. The integrability problem for G-structure and its solution when G is the orthogonal group
4. The Hodge Laplacian and the Bochner trick
The method of moving frames
1. Examples, mainly from from Cartan's book on exterior differential systems, as time permits

Text

There will be no required text, however, I will be using Exterior Differential Systems by Bryant, et al, as my major source. A copy will be placed on reserve in the library, but can also be ordered from the bookstore. I will be supplementing this with other lecture notes that I am writing now.

email address: bryant@math.duke.edu
office 'phone: (919)-660-2805 
office number: 128A Physics