To what extent does the connection one-form on a principal fibre bundle determine the Ehresmann connection?

Question

I have seen a few sources that define the connection on a principal fibre bundle as a choice of "horizontal vector space" $H_pP$ (to my understanding this is an Ehresmann connection) and also as a choice of Lie algebra valued connection 1-form. I have also seen some sources state that there is some relationship between the two (i.e. can determine the other). My question is how uniquely does the connection 1-form determine an Ehresmann connection and vice-versa?

Answer 1

An Ehresmann connection and a connection $1$-form are basically the same thing.

Starting off with an Ehresmann connection, i.e. a "horizontal" subbundle $HP$ of $TP$ complementary to the vertical bundle $VP$, we may also define a projection $TP \to VP$. This projection is just a fiberwise operation coming from basic linear algebra: If $W$ is a vector space with subspace $W'$, then a projection $W \to W'$ is equivalent to a complement $W''$ to $W'$.

Moreover, on a principal bundle, there is a diffeomorphism $P \times \mathfrak{g} \to VP$ given by $(p,X) \mapsto \frac{\rm{d}}{\mathrm{d}t} pe^{tX} |_{t=0}$. So putting the data together, we get a map $TP \to VP \to P \times \mathfrak{g} \to \mathfrak{g}$ where the last map is just projection onto the second factor. This composition is the connection $1$-form. On the other hand, given a connection $1$-form $\omega$, we can define the horizontal subbundle to be the kernel of $\omega$.