We define cubical singular cohomology. We prove the de Rham Theorem, which states that de Rham cohomology is isomorphic to (cubical) singular cohomology.

Read MoreWe explain how horizontal equivariant vector-valued forms on a principal bundle correspond to bundle-valued forms on the associated bundle.

Read MoreWe show how to create associated fibre bundles from a principal bundle. This reproves the vector bundle construction results from earlier lectures.

Read MoreWe define principal bundles. Examples of principal bundles include homogeneous spaces (as the base space) and the frame bundle of a vector bundle.

Read MoreWe prove the global Stokes' Theorem. We then prove that de Rham cohomology is a homotopy invariant, and use this to prove the Poincaré Lemma.

Read MoreWe define a singular cube in a manifold, and explain how to integrate a differential form over a singular cube. We prove a local version of Stokes’ Theorem.

Read MoreWe define topological and smooth manifolds with boundary, and show how an orientation of a smooth manifold with boundary induces one on its boundary.

Read MoreWe prove Cartan's Magic Formula. We then discuss orientability and give three equivalent conditions for a vector bundle to be orientable.

Read MoreWe define differential forms. We construct the exterior differential as a graded derivation of degree one, and prove it commutes with the Lie derivative.

Read MoreWe define tensor fields on manifolds, and prove that the Lie derivative extends to a tensor derivation that commutes with all contractions.

Read MoreWe define (pres)heaves, and give the sheaf-theoretic definition of a manifold. We explain the relation between vector bundles and locally free sheaves.

Read MoreWe prove there is a bijective correspondence between vector bundle homomorphisms and operators on sections that are linear over smooth functions.

Read MoreWe define tensor products, tensor algebras, and the exterior algebra, first for vector spaces, and then for vector bundles.

Read MoreWe prove a “metatheorem” that natural constructions on vector spaces give rise to natural constructions on vector bundles. Oh, and also category theory.

Read MoreWe define fibre bundles. We focus on the case where the fibre is a vector space and the structure group is a matrix Lie group—these are vector bundles.

Read MoreWe define homogeneous spaces, and show that any manifold admitting a transitive Lie group action is a homogeneous space.

Read MoreWe introduce distributions on a manifold, and prove the Frobenius Theorem: a distribution is integrable if and only if it is induced by a foliation.

Read MoreWe define the exponential map of a Lie group, then investigate Lie group actions on manifolds. We conclude by looking at the adjoint representation.

Read MoreWe define Lie groups, and explain why the tangent space to the identity of a Lie group can naturally be seen as a Lie algebra.

Read MoreWe define the flow of a vector field on a manifold. We then define the Lie derivative and explain its relation to the Lie bracket.

Read More