Math 518. Differentiable Manifolds I Instructor Syllabus
The class will meet for three contact hours.
Your grade will be based on homework assignments (25%), one midterm (25%), and a comprehensive final exam (50%).
Topics to be covered:
1. Manifolds: Definitions and examples including projective spaces and Lie groups; smooth functions and mappings; submanifolds; the inverse function theorem and its applications including transversality; (co)tangent vectors and bundles; Whitney's embedding theorem; manifolds with boundary; orientations.
2. Calculus on Manifolds: Vector fields, flows, and the Lie derivative/bracket; differential forms and the exterior algebra of forms; orientations again; exterior derivatives, contraction, and the Lie derivative of forms; integration and Stokes Theorem.
3. Theorem and applications: 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, Dennis Barden and Charles B. Thomas, Imperial College Press, 2003.
Differentiable Topology, Victor Guillemin and Alan Pollack, Prentice Hall, 1974.