The bundle of symmetric affine connections as quotient of the second-order frame bundle

Question

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!

Answer 1

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 $\mathrm{d}\eta=0$.