Handbook

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 an course on differentiable manifolds and related basic concepts, which are the common ground for differential geometry, differential topology, global analysis, i.e. calculus on manifolds including geometric theory of integration, and modern mathematical physics.

Foundational topics covered in the unit include: Smooth manifolds and coordinate systems, tangent and cotangent bundles, tensor bundles, tensor fields and differential forms, Lie derivatives, exterior differentiation, connections, covariant derivatives, curvature, and Stokes's Theorem.

This unit will also cover advanced topics and applications such as: Degree Theory, de Rham cohomology, symplectic geometry, classical mechanics, the Hopf-Rinow theorem, Lie Groups and homogeneous spaces.