Two connections on a smooth vector bundle are called equivalent if they can be mapped to each other by a self-diffeomorphism of the total space that covers identity on the base? Is there a classification of connections up to equivalence?
EDIT: Recall that if $\xi$ is a vector bundle with total space $E$, and if $V$ is the subbundle of the tangent bundle $TE$ whose fibers are the tangent spaces to the fibers of $\xi$, then a connection on $\xi$ is a scale-invariant subbundle $H$ of $TE$ such that $TE=V\oplus H$. Here scale-invariant means that $H$ is preserved by each diffeomorphisms $r:E\to E$ that multiplies a vector by the non-zero real number $r$. From this definition one might suspect that any two connections on $\xi$ are equivalent in the sense of the previous paragraph. Is this true?
The answer to the last question in the edit is NO, as was explained to me by John Etnyre. For example, consider two connections with subbundles $H_1$, $H_2$ such that $H_1$ is integrable everywhere, and $H_2$ is not. Since diffeomorphisms preserve integrability, the two connections aren't equivalent in my sense.
I was hoping to use one of Gromov's h-principles to see when $H_1$ can be deformed to $H_2$ by an ambient isotopy of $E$. Indeed, it is easy to see that any two $H_1$, $H_2$ are homotopic subbundles, e.g. choose Riemannian metric $g_i$ on $E$ such that $H_i$ is $g_i$-orthogonal to the fibers of $E\to B$, and then $g_t=tg_1+(1-t)g_2$ is a Riemannian metric on $E$ and we can let $H_t$ be the orthogonal complement to the fibers; this $H_t$ is the desired homotopy. So one hopes that h-principle can upgrade the homotopy to an isotopy but a simple dimension count shows that we are in the wrong dimension range so h-principle does not apply, as confirmed by the example in part 1.