I've been asked read up on Ehresmann connections. I have experience in smooth manifold theory, vector bundles, and Riemannian geometry, but I feel in unfamiliar territory after briefly looking into the definition online. Therefore, I would like to find a good book so that I can learn the groundwork required to understand what an Ehresmann connection is.
It seems there are a few ways to characterize an Ehresmann connection. My professor describes it as being defined in terms of a horizontal/vertical splitting for locally trivial fibrations, so I'd like to figure out what this means and how the Ehresmann connection is related to this "splitting".
You can find this and more covered in Tu's book Differential Geometry: Curvature, Connections, and Characteristic Classes. Here's the link: https://www.springer.com/gp/book/9783319550824
You'll find this in the later sections of the book, but I would say it's a fairly leisurely read.