Math 641
Introduction to Differential Geometry
Fall 2008

Course Information:

Differential geometry is the application of multivariate calculus to the study of curves, surfaces, and their generalizations to n-dimensional space, smooth manifolds. Ideas and concepts of differential geometry cut across pure and applied mathematics: in physics, mathematical models for mechanical systems, for electromagnetic fields, gravitational fields use the language of differential forms and differential geometry; in optimization, optimal control problems are formulated using ideas of Riemannian and sub-Riemannian geometry; in mathematics, differential geometry plays an essential role in studying the topology and geometry of manifolds.

This course will develop differential geometry beginning with the theory of surfaces in three-dimensional space and continuing with the geometry of Riemannian manifolds in n-dimensions. It is the first part of a two-semester sequence which will fully explore Riemannian differential geometry and some interesting applications, including Hodge theory and spectral geometry (second semester), plus other topics dictated by student interests. It is open to students of mathematics and physics with a good background in linear algebra and multivariate calculus.

The course text for the first semester will be:

Manfredo P. do Carmo. Riemannian geometry. Translated from the second Portuguese edition by Francis Flaherty. Mathematics: Theory and Applications. Birkhäuser Boston, Inc., Boston, MA, 1992

A recommended text is:

Manfredo P. do Carmo, Differential Forms and Applications (Springer-Verlag)

1. Differential Geometry of Surfaces
2. The Gauss-Bonnet Theorem
3. Differential Manifolds
4. Riemannian Metrics
6. Riemannian Connections
7. Geodesics and Curvature

The course requirements will be as follows:

Weekly Problem Sets	60%
Mid-Term Exam	20%
Oral Presentation	20%

Each student will be required to do a final presentation on a topic or short paper in differential geometry. The purpose of this exercise will be not simply to learn an interesting topic in differential geometry but also to gain practice and confidence in communicating mathematical ideas.

References

In addition to the course text, the following are useful references.

1. do Carmo, Manfredo P. Differential geometry of curves and surfaces. Translated from the Portuguese. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976.
2. Lee, John M. Riemannian manifolds. An introduction to curvature. Graduate Texts in Mathematics, 176. Springer-Verlag, New York, 1997.
3. Lee, John M. Introduction to smooth manifolds. Graduate Texts in Mathematics, 218. Springer-Verlag, New York, 2003.
4. Millman, Richard S.; Parker, George D. Elements of differential geometry. Prentice-Hall Inc., Englewood Cliffs, N. J., 1977.
5. Milnor, John. Morse theory. Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51. Princeton University Press, Princeton, N.J. 1963
6. Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques. Riemannian geometry. Third edition. Universitext. Springer-Verlag, Berlin, 2004.
7. Petersen, Peter. Riemannian geometry. Second edition. Graduate Texts in Mathematics, 171. Springer, New York, 2006.