  Let $FM,\mathcal{M}_M$ be the bundles of linear frames and Riemannian metrics
of a manifold $M$, respectively. The existence of a unique
$\mathrm{Diff}M$-invariant connection form on $J^1\mathcal{M}_M\times_MFM\to
J^1\mathcal{M}_M$, which is Riemannian with respect to the universal metric on
$J^1\mathcal{M}_M\times_MTM$, is proved. Aplications to the construction of
universal Pontryagin and Euler forms, are given.
