MATH 6520 - Differentiable Manifolds I

Reyer Sjamaar, fall 2016.

Prerequisites

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

Reference Text

John Lee, Introduction to Smooth Manifolds (most recently)

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. De Rham theory.
3. Transversality.
4. Morse theory.