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.)
- Morse Theory.