MATH 6620

Riemannian Geometry

Course Description

Course information provided by the 2024-2025 Catalog.

Topics include linear connections, Riemannian metrics and parallel translation; covariant differentiation and curvature tensors; the exponential map, the Gauss Lemma and completeness of the metric; isometries and space forms, Jacobi fields and the theorem of Cartan-Hadamard; the first and second variation formulas; the index form of Morse and the theorem of Bonnet-Myers; the Rauch, Hessian, and Laplacian comparison theorems; the Morse index theorem; the conjugate and cut loci; and submanifolds and the Second Fundamental form.

Prerequisites/Corequisites Prerequisite: MATH 6520 or strong performance in analysis (e.g., MATH 4130 and/or MATH 4140), linear algebra (e.g., MATH 4310), and coursework on manifolds and differential geometry at the undergraduate level, such as both MATH 3210 and MATH 4540.

When Offered Spring.

Comments Offered alternate years.

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