Course Outline for Math 240A (Fall 2011)

Differential Geometry

MWF 9 am, Girvetz 2124

Instructor: John Douglas Moore

The first quarter of the differential geometry course will be devoted to global calculus of several variables, also known as calculus on manifolds.

The simplest examples of manifolds are surfaces in Euclidean space, and in fact manifolds can be thought of as simply n-dimensional generalizations of such surfaces. Manifolds are necessary for the formulation of global problems in geometry, analysis and physics. They form the foundation for the three great theories of modern mathematics, differential equations, differential topology and differential geometry.

Topics to be treated in 240A include tangent vectors, vector bundles, submanifolds, tensors, differential forms and the manifold version of Stokes’ Theorem. This will lead us up to de Rham’s beautiful cohomology theory, which measures the number of holes of various dimensions in a manifold.

The second and third quarters of the continuing course (240BC) will treat Riemannian geometry, Lie groups and the theory of connections.

We plan to provide brief lecture notes for the course, which will include the homework problems.

The official text for the course is: John M. Lee, Introduction to Smooth Manifolds, Springer, 2003. Since I will not follow the text very closely it is not absolutely necessary to buy the text. In fact, the bookstore was not able to get copies from the publisher, but the book appears to be available from Amazon.com. The Introduction and Chapter 1 are available directly from Lee's website at:

http://www.math.washington.edu/~lee/Books/smooth.html

That should give you time to order the text from Amazon.com or some other source if you want to do that.  You can probably find a pdf of earlier versions of the text on the web.