MATH 6520 Differentiable Manifolds I

Prerequisites

Undergraduate analysis, linear algebra, and point-set topology.

Minimum Syllabus

1. Manifolds, submanifolds. Immersions, embeddings and submersions.
2. Tangent bundles and tangent maps. Vector fields, derivations and the Lie bracket.
3. Sard’s theorem, easy Whitney embedding theorem.
4. Trajectories and flows of vector fields. Frobenius integrability theorem.
5. Connections, curvature and geodesics. Riemannian metrics, Levi-Civita connections.
6. Tensors, differential forms. Exterior derivative and Stokes’ theorem.

Optional Topics

1. Lie groups, Lie algebras, homogeneous spaces. 
2. Classification of 1- and 2-manifolds.
3. De Rham theory. (Requires some elementary homological algebra — snake Lemma, five lemma — which should be stated without proof.)
4. Transversality.
5. Morse Theory.