Stäckel and Eisenhart lifts, Haantjes geometry and Gravitation

Abstract: We introduce the notion of St\"ackel lift as a novel geometric setting for the construction of large classes of integrable Hamiltonian systems. The St\"ackel lift extends the geometric framework underlying both the Riemannian and the Lorentzian-type classical Eisenhart lifts and is intimately related with Haantjes geometry. In particular, we establish that Hamiltonian systems constructed through St\"ackel lifts are naturally endowed with a symplectic-Haantjes structure. We also show that explicitly momentum-dependent lifting matrices produce systems interpretable as gravitational waves, or momentum-dependent metrics of Hamilton and Finsler geometries, with potential applications in modified gravity theories.