Natural connections on the bundle of Riemannian metrics

Abstract

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 $J^1{\mathcal{M}}_M{\times }_{M}FM\to J^1{\mathcal{M}}_M$, which is Riemannian with respect to the universal metric on $J^1{\mathcal{M}}_M{\times }_{M}TM$, is proved. Aplications to the construction of universal Pontryagin and Euler forms, are given.