Differential geometry

Manifolds are topological spaces that are locally homeomorphic to Euclidean space. A differentiable structure on a manifold makes it possible to generalize many concepts from calculus in Euclidean spaces to manifolds. This is a course on differentiable manifolds and related basic concepts, which are the common ground for differential geometry, differential topology, and geometric analysis.

Foundational topics covered in the unit include: Smooth manifolds and coordinate systems, submanifolds, tangent and cotangent bundles, tensor bundles, tensor fields, Lie derivatives and tensor derivations.

This unit will also cover advanced topics and applications such as: degree theory, de Rham cohomology, symplectic geometry  and classical mechanics, Riemmanian geometry, comparison geometry, Lie groups and homogeneous spaces.