*This post is not about finding an answer to a certain problem - because the answer already exists - but rather about finding the simplest possible answer.*


**The problem is**: how to define the bundle $C(M)$ whose sections are all the symmetric affine connections on the manifold $M$ in *terms of natural bundles over $M$* in the simplest possible way?


If we drop the hypotesis of symmetricity, then the answer probably is
$$
J^1(F(M))/GL\_n(\mathbb{R})\quad\quad (\*)
$$
where $F(M)$ is the frame bundle. I believe that formula $(\*)$ actually becomes simpler *with* the hypotesis of symmetricity.


My idea goes like this. Consider the following bundle
$$
\check{J}^k(M,\mathbb{R}\_0^n):=\{[\boldsymbol{x}]\_{m,0}^k\mid\boldsymbol{x}\textrm{ is a local diffeomorphism sending }m\in M\textrm{ to }0\in \mathbb{R}^n\}
$$
of $k$-jets of local diffeomorphisms from $M$ to $\mathbb{R}^n$ together with its natural projection on $M$ (like in P. Michor's book about manifolds of differentiable mappings). In particular,
$$
\check{J}^1(M,\mathbb{R}\_0^n)=F(M)
$$
is a $GL\_n(\mathbb{R})$-principal bundle and each $\check{J}^k(M,\mathbb{R}\_0^n)$ carries a natural $GL\_n(\mathbb{R})$-action. I'm sure that the sub-bundle of $(\*)$ made of symmetric affine connection is precisely
$$
C(M)=\check{J}^2(M,\mathbb{R}\_0^n)/GL\_n(\mathbb{R})\quad\quad (\*\*)
$$ 



> 
> **QUESTION:** is the identification $(\*\*)$ correct? If yes, is there a direct proof of it, i.e., not passing through $(\*)$? Can $(\*\*)$ be found, formalised exactly as above, in the literature?
> 
> 
> 


Needless to say, either if my guess $(\*\*)$ is wrong or if there are better answers out there, I'd like to know!



The bundle of symmetric affine connections as quotient of the second-order frame bundle
There *is* an 'identification', i.e., a way to interpret a torsion-free affine connection on $M$ as a section of $\check J^2(M,\mathbb{R}^n\_0)/\mathrm{GL}\_n(\mathbb{R})$ in such a way that every (smooth) section of $\check J^2(M,\mathbb{R}^n\_0)/\mathrm{GL}\_n(\mathbb{R})$ over $M$ corresponds to a unique torsion-free (smooth) affine connection. You haven't explicitly *described* such an identification, though, so it's hard to say whether $(\*\*)$ is 'correct'.


Here is one way to do it: Given a torsion-free connection $\nabla$ on an $n$-manifold $M$, and a point $m\in M$ let $\exp\_m^\nabla:T\_mM\to M$ be the (locally defined) exponential map, which is a diffeomorphism from an open neighborhood of $0\_m\in T\_mM$ to an open $m$-neighborhood $U\_m\subset M$. Choose a linear isomorphism $u: T\_mM\to\mathbb{R}^n$ and let $x = u^{-1}\circ (\exp\_m^\nabla)^{-1}:(U\_m,m)\to(\mathbb{R}^n,0)$. Then the $\mathrm{GL}\_n(\mathbb{R})$ equivalence class $[x]^2\_{m,0}{\cdot} \mathrm{GL}\_n(\mathbb{R})$ is a canonically-determined element $\gamma(\nabla)\_m$ in the $m$-fiber of the bundle $\check J^2(M,\mathbb{R}^n\_0)/\mathrm{GL}\_n(\mathbb{R})\to M$. Thus, $\nabla$ canonically determines a section $\gamma(\nabla)$ of $\check J^2(M,\mathbb{R}^n\_0)/\mathrm{GL}\_n(\mathbb{R})\to M$. 


It's easy to check (in local coordinates) that this mapping from connections to sections of the given bundle has all of the desired properties. In particular, every smooth section $\sigma$ of $\check J^2(M,\mathbb{R}^n\_0)/\mathrm{GL}\_n(\mathbb{R})\to M$ is of the form $\sigma = \gamma(\nabla)$ for some unique smooth torsion-free affine connection on $M$.


Note that the natural affine structure on $\check J^2(M,\mathbb{R}^n\_0)/\mathrm{GL}\_n(\mathbb{R})\to M$ is described as follows: Let $[x]^2\_{m,0}{\cdot}\mathrm{GL}\_n(\mathbb{R})$ and $[y]^2\_{m,0}{\cdot}\mathrm{GL}\_n(\mathbb{R})$ be two elements in the $m$-fiber and suppose that two local coordinate representatives have been chosen so that $[x]^1\_{m,0}=[y]^1\_{m,0}$. Then define 
$$
t\,\bigl([x]^2\_{m,0}{\cdot}\mathrm{GL}\_n(\mathbb{R})\bigr)+(1{-}t)\,\bigl([y]^2\_{m,0}{\cdot}\mathrm{GL}\_n(\mathbb{R})\bigr)
= \bigl([t\,x + (1{-}t)\,y]^2\_{m,0}{\cdot}\mathrm{GL}\_n(\mathbb{R})\bigr).
$$
One can check (again, in local coordinates), that this is a smooth affine action on $\check J^2(M,\mathbb{R}^n\_0)/\mathrm{GL}\_n(\mathbb{R})\to M$ and that 
$$
\gamma(t\,\nabla\_1 + (1{-}t)\,\nabla\_2) = t\,\gamma(\nabla\_1) + (1{-}t)\,\gamma(\nabla\_2).
$$


As far as references go, I don't know for sure, but I would not be at all surprised to find that this precise construction is described somewhere in Charles Ehresmann's original papers on natural jet bundles.


**Added remark:** One way to make this match a little better with the description of the general affine connection is to use, instead, the *coframe* bundle $F^\*(M)\to M$, where a coframe $u\in F^\*(M)$ is a linear isomorphism $u:T\_mM\to\mathbb{R}^n$. A local section of $F^\*(M)$ is just a *coframing*, i.e., an $\mathbb{R}^n$-valued $1$-form $\eta$ on $U\subset M$ such that $\eta\_m:T\_mU\to\mathbb{R}^n$ is an isomorphism for all $m\in U$. Then, in a natural way, the space of affine connections is identified with the sections of $J^1(F^\*(M))/\mathrm{GL}\_n(\mathbb{R})$, while the space of torsion-free affine connections is identified with the space of sections of the $\mathrm{GL}\_n(\mathbb{R})$-quotient of the submanifold $J^1\_0(F^\*(M))\subset J^1(F^\*(M))$ consisting of $1$-jets of *closed* coframings, i.e., the local coframings $\eta$ that satisfy $\mathrm{d}\eta=0$.


