**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.