Affine connections as equivariant maps

Question

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.

Answer 1

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 $P = FM$.

Answer 2

I am not sure of your definition of second order coframe bundle. If we say that the second order coframe bundle is defined to be the set of all choices of a point of the coframe bundle and a torsion-free pseudoconnection 1-form at that point, then the structure group of the second order coframe bundle as a bundle over the first order coframe bundle is exactly your $S^2 \mathbb{R}^{n*} \otimes \mathbb{R}^n$: any two such pseudoconnections differ by an element of the first prolongation of the general linear group. If instead we define the second order coframe bundle to be the set of all 2-jets of local coordinates, then those with a given 1-jet (i.e. over the same point of the coframe bundle) agree up to arbitrary quadratic terms, i.e. structure group $S^2 \mathbb{R}^{n*} \otimes \mathbb{R}^n$. The terminology in formal differential geometry (or whatever this sort of thing is called) is not standardized much.

A torsion-free connection is a section of $FM^1 \to FM$, equivariant for the general linear group. So I don't think you really don't get an action of the structure group of $FM^1 \to M$ on $S^2 \mathbb{R}^{n*} \otimes \mathbb{R}^n$ arising in this way: you are not making a difference between connections when you act by the structure group; you are making a pseudoconnection form move to a different point of the coframe bundle. If you fix that point of the coframe bundle, you get what I wrote in the last paragraph.