# 28. Connections

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.

Comments and questions?