Math 520: Differentiable Manifolds I

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: Sard's Theorem, Distributions and the Frobenius Theorem; intersection theory and degree; Lefschetz Fixed Point Theorem; Poincare-Hopf Index Theorem; DeRham cohomology.


An Introduction to Differential Manifolds (Paperback) by Dennis Barden and Charles B. Thomas, Imperial College Press; Reprint edition (March 2003).


The course grade will be based on weekly homework (35%), a midterm (25%) and a final (40%).
lecture of 11/16: Mayer-Vietoris

Last modified: Mon Nov 26 15:06:06 CST 2007