Math 518: Differentiable Manifolds I
Fall 2011

Basic Information


Point set topology and linear algebra will be very useful.

If you have any questions or concerns, please contact me by e-mail.

Course outline

  1. Manifolds: Definitions and examples including projective spaces and Lie groups; smooth functions and mappings; submanifolds; Inverse Function Theorem and its applications including transversality; (co)tangent vectors and bundles; Whitney Embedding Theorem; manifolds with boundary; orientations.

  2. Calculus on Manifolds: Vector fields, flows, and Lie derivative/bracket; differential forms and the exterior algebra of forms; orientations again; exterior derivative, contraction, and Lie derivative of forms; integration and Stokes Theorem.

  3. Other topics (depending on how much time we'll have): Sard's Theorem, Distributions and the Frobenius Theorem; Lie groups; DeRham cohomology.


Introduction to Smooth Manifolds by John M. Lee may be useful. It is not required.

I will mostly follow these notes. Their state is not final and they will be edited over the course of the semester. So don't print out the whole file.
If your point set topology is rusty, have a look at this short file.


The course grade will be based on weekly homework and a final exam. If there is an overwhelming popular demand, a midterm may be arranged.

Homework assignments

If you want to learn more geometry, next semester you could take 519, where we'll be doing connections, curvature and characteristic classes. And you could take 524 .

Notes of lectures

Last modified: Wed Dec 7 14:04:54 CST 2011