Natural connections on the bundle of Riemannian metrics

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