Math 641
Introduction to Differential Geometry
Fall 2008

Instructor Peter A. Perry, 755 POT
Office755 Patterson Office Tower
Phone257-6791
E-Mailperry@ms.uky.edu
Office HoursWF 4:00-5:00 PM
Class MeetingsMWF 3:00-3:50 CB 349

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 oF Riemannian manifolds in n-dimension 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

During the first semester, we will cover the following topics:

  1. Review of Multivariate Calculus
  2. Differential Manifolds
  3. Riemannian Metrics
  4. Riemannian Connections
  5. Geodesics and Curvature

The course requirements will be as follows:

Problem Sets60%
Mid-Term Exam20%
Final Problem Set 20%

Lecture Notes

Homework and Handouts

Exams

References

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

  1. Manfredo P. do Carmo, Differential Forms and Applications (Springer-Verlag)
  2. do Carmo, Manfredo P. Differential geometry of curves and surfaces. Translated from the Portuguese. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976.
  3. Lee, John M. Riemannian manifolds. An introduction to curvature. Graduate Texts in Mathematics, 176. Springer-Verlag, New York, 1997.
  4. Lee, John M. Introduction to smooth manifolds. Graduate Texts in Mathematics, 218. Springer-Verlag, New York, 2003.
  5. Millman, Richard S.; Parker, George D. Elements of differential geometry. Prentice-Hall Inc., Englewood Cliffs, N. J., 1977.
  6. 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
  7. Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques. Riemannian geometry. Third edition. Universitext. Springer-Verlag, Berlin, 2004.
  8. Petersen, Peter. Riemannian geometry. Second edition. Graduate Texts in Mathematics, 171. Springer, New York, 2006.
  9. Spivak, Michael. Calculus on manifolds. A modern approach to classical theorems of advanced calculus. W. A. Benjamin, Inc., New York-Amsterdam 1965

And, of course, the monumental five-volume treatise by Spivak on Differential Geometry!