Math 520. Differentiable Manifolds

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Math 520. Differentiable Manifolds

Required Text (for Fall 2000 Semester):
Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd Ed, Academic Press

Recommended Texts (for Fall 2000 Semester):
Morgan, Riemannian Geometry: A Beginner's Guide, 2nd Ed, A K Peters
Bishop and Goldberg, Tensor Analysis on Manifolds, Dover
Milnor, Topology from the Differentiable Viewpoint, U P Virginia
Spivak, Calculus on Manifolds, Benjamin/Cummings

Outline:
This is a first course in manifolds and global analysis, which will present the basic tools for those interested in, or curious about, differential geometry or global analysis, or those who want to apply differential geometric methods in other areas such as PDE, topology, mathematical physics, and dynamical systems.

The course will cover the following topics:

Manifolds:
- Definition and examples of differentiable manifolds. Partition of unity. Differentiable functions and mappings. Implicit function theorem and rank theorem. Immersions and imbeddings. Submanifolds.
- Tangent vectors, Tangent bundles.
Calculus on manifolds:
- Vector fields. Integral curves and flows. Lie bracket.
- Differential forms, exterior calculus and algebra of differential forms. Closed and exact forms. Poincare Lemma, DeRham complex.
- Oriented manifolds, equivalent definitions of orientability.
- Integration of differential forms. Stokes's Theorem.
Connections:
- Connections on vector bundles
- Parallel transport
- Levi-Civita connection

Graduate Guide homepage

Department of Mathematics
University of Illinois at Urbana-Champaign
273 Altgeld Hall, MC-382
1409 W. Green Street, Urbana, IL 61801 USA
Telephone: (217) 333-3350 Fax (217) 333-9576
office@math.uiuc.edu

Revised 11/21/00
Reviewed by Area Chair 6/20/05