Diffeomorphism-invariant covariant Hamiltonians of a pseudo-Riemannian metric and a linear connection. (English) Zbl 1266.53027

Summary: 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 \(M\times_NC\) is determined such that, for every connection \(\gamma\) belonging to this class and every \(\operatorname{Diff}(N)\)-invariant Lagrangian density \(\Lambda \) on the jet bundle \(J^1(M\times _NC)\), the corresponding covariant Hamiltonian \(\Lambda ^\gamma \) is also \(\operatorname{Diff}(N)\)-invariant. The case of \(\operatorname{Diff}(N)\)-invariant second-order Lagrangian densities on \(J^2M\) is also studied and the results obtained are then applied to Palatini and Einstein-Hilbert Lagrangians.


53C05 Connections (general theory)
58E30 Variational principles in infinite-dimensional spaces
58A20 Jets in global analysis
58J70 Invariance and symmetry properties for PDEs on manifolds
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
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