Course Outline for Math 240BC (Winter-Spring 2007)

Differential Geometry

PLEASE NOTE TIME AND ROOM CHANGE:  MWF 12 noon, SH 4519

Instructor: J. Douglas Moore

Prerequisites: 240A or 221C or consent of instructor

Tentative Outline of the Course:   Roughly speaking, differential geometry is the application of ideas from calculus (or from analysis) to geometry. It has important connections with topology, partial differential equations and a subtopic within differential geometry---Riemannian geometry---is the mathematical foundation for general relativity. Another branch of differential geometry, connections on fiber bundles, is used in the standard model for particle physics.

This course will describe the foundations of Riemannian geometry, including geodesics and curvature, as well as connections in vector bundles, and then go on to discuss the relationships between curvature and topology.   Topology will presented in two dual contrasting forms, de Rham cohomology and Morse homology.  To provide background for the second idea, we will describe some of the calculus of variations in the large originally developed by Marston Morse.  This theory shows, for example, that many Riemannian manifolds have many geometrically distinct smooth closed geodesics.  If time permits, we may give a brief mathematical introduction to general relativity, one of the primary applications.

Recommended References:   We will develop lecture notes for the course.  However, there are many excellent texts that can help supplement the notes, including:

1.     William M. Boothby, An Introduction to Differentiable Manifolds and Lie Groups, Second Edition, Academic Press, New York, 2003.  (The first four chapters of this text were discussed in Math 240A.  Math 231C also presents manifold theory.)

2.     Manfredo P. do Carmo, Riemannian Geometry, Birkhauser, Boston, 1992.  This is one of the standard references on the topic.

3.     John M. Lee, Riemannian Manifolds, Springer, 1997.  A short readable overview.

4.     Jurgen Jost, Riemannian Geometry and Geometric Analysis, Fifth Edition, Springer, 2008.  Contains much more than can be discussed in the course.  One of the few book treatments of Morse homology.

5.     John Milnor, Morse Theory, Princeton University Press, Princeton, 1969.  The classic treatment of the topology of critical points of smooth functions on manifolds.