### 🎉🎉🎉 Welcome to Differential Geometry II 🎉🎉🎉

I hope you all enjoyed your vacation.

I didn’t.

I spent the entire month doing exams.

Anyway…  🙃

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!