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

Abstract

\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 $\mathrm{Diff}N$-invariant Lagrangian density $\Lambda$ on $J^1(M{\times }_{N}C)$, the corresponding covariant Hamiltonian ${\Lambda }^{\gamma }$ is also $\mathrm{Diff}N$-invariant. The case of $\mathrm{Diff}N$-invariant second-order Lagrangian densities on $J^{2}M$ is also studied and the results obtained are then applied to Palatini and Einstein-Hilbert Lagrangians.

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