Differential Geometry I introduced the basics of smooth manifolds and bundle theory. Differential Geometry II will primarily be concerned with two extra pieces of data one can endow a manifold or bundle with: a connection and a Riemannian metric. The study of connections on bundles is usually called gauge theory, and the study of Riemannian manifolds — that is, smooth manifolds equipped with a Riemannian metric — is referred to as Riemannian geometry.

We begin with the former.

A preconnection on a fibre bundle $\pi \colon E \to M$ is a distribution $\Delta$ on $E$ complementary to the vertical bundle. In the special case where $E$ is itself a vector bundle, we define a connection as a preconnection that respects the scalar multiplication action.

Next lecture we will introduce parallel transport, which gives another way to define connections. After that we will show that a connection is equivalent to a covariant derivative $\nabla_X$, which is perhaps the point of view most commonly taken. (This is what will eventually lead to the Christoffel symbols $\Gamma^i_{ij}$ some of you may have seen before). Eventually we will unify all these approaches to connections by studying connections on principal bundles.