MATH 6520: Differentiable Manifolds I (fall 2008)

Instructor: Ed Swartz

We shall develop (many of) the basic theorems and apparatus used in their study. This will include inverse and implicit function theorems in Euclidean spaces, definition and examples of differrentiable manifolds, tangent vectors and bundles, functorial passage from vector space constructions to bundle constructions, vector fields, ordinary differential equations on manifolds, Lie bracket, Lie derivative and Frobenius' Theorem, Lie groups and their Lie algebras, Sard's Theorem, Embedding in Euclidean spaces, tensor algebra, tensor fields, exterior derivative, integration of differential forms, Stokes' Theorem, De Rham cohomology groups and applications.

We shall use both Boothby's An Introduction to Differentiable Manifolds and Differential Geometry, and Conlon's Differentiable Manifolds.