MATH 6520

Differentiable Manifolds

Course Description

Course information provided by the Courses of Study 2022-2023.

MATH 6510-MATH 6520 are the core topology courses in the mathematics graduate program. MATH 6520 is an introduction to geometry and topology from a differentiable viewpoint, suitable for beginning graduate students. The objects of study are manifolds and differentiable maps. The collection of all tangent vectors to a manifold forms the tangent bundle, and a section of the tangent bundle is a vector field. Alternatively, vector fields can be viewed as first-order differential operators. Students study flows of vector fields and prove the Frobenius integrability theorem. In the presence of a Riemannian metric, the notions of parallel transport, curvature, and geodesics are development. Students examine the tensor calculus and the exterior differential calculus and prove Stokes' theorem. If time permits, de Rham cohomology, Morse theory, or other optional topics are introduced.

When Offered Fall.

Prerequisites/Corequisites Prerequisite: strong performance in analysis (e.g., MATH 4130 and/or MATH 4140), linear algebra (e.g., MATH 4310), and point-set topology (e.g., MATH 4530), or permission of instructor.

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