Diffeomorphism-invariant Covariant Hamiltonians of a pseudo-Riemannian Metric and a Linear Connection

\noindent Let $M\to N$ (resp.\ $C\to N$) be the fibre bundle of pseudo-Riemannian metrics of a given signature (resp.\ the bundle of linear connections) on an orientable connected manifold $N$. A geometrically defined class of first-order Ehresmann connections on the product fibre bundle $M{\times }_{N}C$ is determined such that, for every connection $\gamma$ belonging to this class and every $\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}N$-invariant Lagrangian density $\mathrm{\Lambda }$ on $J^1(M{\times }_{N}C)$, the corresponding covariant Hamiltonian $\Lambda ^\gamma $isalso$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}$N$-invariant\mathrm{.}Thecaseof$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}$N$-invariantsecond-orderLagrangiandensitieson$J^2M$ is also studied and the results obtained are then applied to Palatini and Einstein-Hilbert Lagrangians.