An affine torsion-free connection on a smooth manifold $M$ may be thought of as a section of an affine bundle whose associated vector bundle is $S^2(T^\*M)\otimes TM$. One would think that this affine bundle is an associated bundle to the second order (co-)frame bundle $F^2(M) \to M$. Hence there should be a natural affine action $\chi$ of the structure group of $F^2(M)$ on $S^2(\mathbb{R}\_n)\otimes \mathbb{R}^n$, so that affine connections on $M$ correspond to $\chi$-equivariant maps $f : F^2(M) \to S^2(\mathbb{R}\_n)\otimes \mathbb{R}^n$. I wonder how this action looks like and how it can be derived.



Affine connections as equivariant maps
There is a construction that works for connections on arbitrary principal bundles (and not just for the frame bundle):
Let $P$ be a principal $G$-bundle over a $n$-dimensional manifold $M$. The space of principal connections on $P$ can be identified with sections of an affine bundle $QP \to M$. In order to realize $QP$ as an associated bundle, we need a first-order jet bundle of $P$ (since connections are first-order operators). However, the first-order jet bundle $J^1 P$ fails to be a principal bundle in general and has to be replaced with the so-called first principal jet prolongation $W^1 P \to M$ of $P$. Then $QP$ can be seen as an associated bundle to $W^1 P$ with typical fibre $\mathfrak{g} \times (\mathbb{R}^n)^\*$. The action of the structure group $W^1 G$ of $W^1 P$ on this fibre is a bit complicated and so I've to refer you to section 17.5 of Natural Operations in Differential Geometry
by Kolar, Michor & Jan Slovak for a explicit expression. I'm not sure if these formulas simplify for the special case $P = FM$.


