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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".

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    $\begingroup$ I've seen some good chunk of that in here. $\endgroup$ May 19 '21 at 2:46
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    $\begingroup$ I have some short notes on it, see section 3 here. $\endgroup$
    – Ivo Terek
    May 20 '21 at 6:21
  • $\begingroup$ @IvoTerek thanks a lot, these look like they start in a place I am familiar with. I will give these a read. $\endgroup$
    – epsilon
    May 20 '21 at 7:08
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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.

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