Metric, torsion free connections on principal bundles I hope this is not too elementary, but I have asked this question at the math.stack site, but I have obtained no answers.
Let $F(M)$ be the frame bundle of a $n$-dimensional differentiable manifold $M$, and let $H\subset TF(M)$ be a connection. A Riemannian metric on $M$ can be equivalently written as an equivariant function $f\colon F(M)\to S^2(\mathbb{R}^{n})$. How can one impose the metric compatibility and the torsion free condition on $H$ without using at all the picture on the tangent bundle $TM$?
Thanks.
 A: I don't know what you mean by 'without using at all the picture on the tangent bundle $TM$', but here is how one normally does it:
First, one shows that there are canonical $1$-forms $\omega^i$ on $F(M)$ that are semibasic for the projection $\pi:F(M)\to M$ and that satisfy the equation
$$
\pi'(v) = \omega^i(v)\,e_i(f)
$$
for all $v\in T_fF(M)$, where $f = (e_1,\ldots,e_n)\in F(M)$ is a frame.  
Second one sees that the horizontal plane field $H$ associated to the Levi-Civita connection is characterized as the kernel of the unique (connection) $1$-forms $\theta^i_j$ on $F(M)$ that satisfy
$$
\mathrm{d}g_{ij} = g_{ik}\,\theta^k_j + g_{kj}\,\theta^k_i
$$
where $g_{ij}(f) = \langle e_i,e_j\rangle$ (this is metric compatibility) and
$$
\mathrm{d}\omega^i = -\theta^i_j\wedge\omega^j
$$
(this is the torsion-free condition).  These are known as the (first) structure equations of Élie Cartan for the Levi-Civita connection.
Addendum:  (Added at the request of the OP)  The main point is that, for any connection $1$-form $\phi = (\phi^i_j)$ on $F(M)$, one will have equations of the form
$$
\mathrm{d}g_{ij} = g_{ik}\,\phi^k_j + g_{kj}\,\phi^k_i + g_{ijk}\,\omega^k
$$
and 
$$
\mathrm{d}\omega^i = -\phi^i_j\wedge\omega^j + \tfrac12h^i_{jk}\,\omega^j\wedge\omega^k
$$ 
for some functions on $F(M)$ satisfying $g_{ijk} = g_{jik}$ and $h^i_{jk}=-h^i_{kj}$.  (These are just consequences of the $\mathrm{GL}(n,\mathbb{R})$-equivariance of the matrix $g = (g_{ij})$ and the tautological definition of the $\omega^i$.)  Now, any other connection $\theta = (\theta^i_j)$ will be expressible in the form
$$
\theta^i_j = \phi^i_j + p^i_{jk}\,\omega^k
$$
for some functions $p^i_{jk}$ on $F(M)$.  Then Cartan's structure equations for the Levi-Civita connection $\theta$ as listed above simply become the inhomogeneous linear system
$$
g_{il}\,p^l_{jk} + g_{jl}\,p^l_{ik} = g_{ijk}
\quad\text{and}\quad
p^i_{jk} - p^i_{kj} = h^i_{jk},
$$
which, by the stated index symmetries, is the same number of independent equations as unknowns $p^{i}_{jk}$, namely $n^3$.  By the usual symmetry/antisymmetry argument, the homogeneous equations have the unique solution $p^i_{jk} = 0$, so the inhomogeneous equations have a solution and it is unique.  (Uniqueness implies the needed equivariance, so that the resulting $\theta$ does, indeed define a connection on $F(M)$.) 
