Geometric Intuition for Connection Map Let $ V \to E \xrightarrow{\pi} M$ be a vector bundle with connection $\nabla$. Denote by $K \colon TE \to E$ the corresponding connection map.
What geometric intuition is there for this map? Is there an explicit formula for the connection map of the Levi-Civita connection?
 A: Given $K$, you can "enlarge" the target space in a natural way to the vertical space $VE$, so that you have a mapping $\tilde{K}:TE\to VE$. Now, given $\xi\in TE$, as the name suggests, $\tilde{K}(\xi)$ should be thought of as the "vertical part of $\xi$", and then
\begin{align}
\eta:=(\text{id}_{TE}-\tilde{K})(\xi)=\xi-\tilde{K}(\xi)
\end{align}
is to be thought of as the "horizontal part of $\xi$", and this is precisely what "connects" the various fibers of $E$ on an "infinitesimal level". If you now take a look at this question of mine, I've drawn a picture at the bottom which I think you'll find helpful (the $\eta$ here is what I refer to there as $C(k_x,u_x)$). Thus far, this $\tilde{K}$ (or $\text{id}_{TE}-\tilde{K}$, or equivalently what I called $C$ in my question) are "infinitesimal descriptions", i.e they're working a the level of tangent spaces of $E$. On the other hand, by solving a certain linear ODE, one can actually connect different fibers of $E$, and this is what we call the parallel transport map.
