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College of Arts and Sciences
Department of Mathematics
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MATH 6520 Differentiable Manifolds I
Undergraduate analysis, linear algebra, and point-set topology.
Manifolds, submanifolds. Immersions, embeddings and submersions.
Tangent bundles and tangent maps. Vector fields, derivations and the Lie bracket.
Sard’s theorem, easy Whitney embedding theorem.
Trajectories and flows of vector fields. Frobenius integrability theorem.
Connections, curvature and geodesics. Riemannian metrics, Levi-Civita connections.
Tensors, differential forms. Exterior derivative and Stokes’ theorem.
Lie groups, Lie algebras, homogeneous spaces.
Classification of 1- and 2-manifolds.
De Rham theory. (Requires some elementary homological algebra — snake Lemma, five lemma — which should be stated without proof.)