# 28. Connections on vector bundles

### πππ Welcome to Differential Geometry II πππ

I hope you all enjoyed your vacation.

I didnβt.

I spent the *entire* month doing exams.

Anywayβ¦ Β π

We begin this semester with an extensive discussion of *connections*** **on vector bundles. This will take approximately the first ten lectures.

After a short introductory section, we define a *preconnection* on a fibre bundle $ \pi \colon E \to M$ as distribution 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.

Anyway, here are today's notes. As usual, I'm sure they're full of typos, so please do leave comments on my forum (follow the link below) to let me know!

Comments and questions?