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

Muños Masqué, Jaime; Rosado María, María Eugenia

Adv. Theor. Math. Phys. 16, No. 3, 851-886 (2012).

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.

MSC:

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)

Keywords:

bundle of pseudo-Riemannian metrics; bundle of linear connections; invariant Lagrangian density; second-order Lagrangian densities