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Math 518. Differentiable Manifolds I
Meeting Time:
The class will meet for three contact hours.Grades:
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.
Recommended References
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.
| Department
of Mathematics 273 Altgeld Hall, MC-382 1409 W. Green Street, Urbana, IL 61801 USA Telephone: (217) 333-3350 Fax: (217) 333-9576 Email: office@math.uiuc.edu |
College of Liberal Arts and
Sciences Last modified July 2, 2008 |