Math 620

MATH 620: Smooth Manifolds

Description: This introductory course will cover the topics listed below, plus additional topics (such as principal bundles) as time allows:

1. Basic constructions: smooth manifolds and maps, tangent and cotangent bundles, vector fields, differential forms, orientations, Lie bracket.
2. Smooth maps: embeddings, immersions, submersions, Inverse Function Theorem, Rank Theorem, submanifolds.
3. Lie groups and their Lie algebras, quotients of Lie group actions, Maurer-Cartan forms.
4. Integration on manifolds, Stokes' Theorem.
5. Flows of vector fields, foliations, Frobenius Theorem.
6. Additional topics as time allows.

Prerequisites: Students taking Math 620 are required to have taken real analysis (at the level of Math 531) and are encouraged to have studied point-set topology (at the level of Math 411). They are expected to be familiar with the following concepts: differentiation and integration in $\mathbb{R}^n$, topological spaces, continuous maps, subspace topology, compactness.

Instructor: Robert L. Bryant