Self-dual metrics with maximally superintegrable geodesic flows

Abstract

A class of self-dual and geodesically complete spacetimes with maximally superintegrable geodesic flows is constructed by applying the Eisenhart lift to mechanics in pseudo-Euclidean spacetime of signature (1, 1). It is characterized by the presence of a second rank Killing tensor. Spacetimes of the ultrahyperbolic signature (2, q) with q > 2, which admit a second rank Killing tensor and possess superintegrable geodesic flows, are built.

Null lifts and projective dynamics

“…One question that comes to mind is if one can give a projective interpretation of the phenomenon of R-separation of the Schrödinger equation [25,26,33]. We also notice how the results discussed here imply that recent efforts in obtaining new non-trivial Einstein spaces from dynamical systems [8,19] can be extended to the more general case of Einstein-Weyl spaces, see [6]. A considerable amount of research has been produced in the area, as a non-exhaustive example see [4,28,29].…”

“…This is the equation of a projective conic on PT M, so we can think of the choice of H as a point by point choice of projective conics, or a section in a bundle of projective conics with base M. Because of the projective nature of the conic, a different choice of metric ḡ = Ω 2 (y)g induces the same conic, and therefore what is relevant is the conformal class of [g] and an exact Weyl geometry is naturally induced: one can start with a reference metric g ∈ [g] and its Levi-Civita covariant derivative, which implies ϕ = 0 in this gauge, and then transform g and ϕ according to (2. 19), (2.20). By exact geometry we mean that in every gauge ϕ A dy A is an exact form.…”

Eisenhart lift for higher derivative systems

“…The geodesic equations associated with (32) do reproduce (29), while the evolution of s is fixed provided the general solution of (29) is known. For earlier application of the Eisenhart lift to mechanics in pseudo-Euclidean space see [11,12]. Turning to the odd values of n = 2m + 1, the condition that the function x (n) V + U depends on x and its derivatives of even order only implies…”

“…In a series of recent works [5]- [10] various Lorentzian spacetimes admitting irreducible Killing tensors of rank greater than two have been constructed by applying the Eisenhart lift to specific integrable models. In [11,12] Ricci-flat spacetimes of the ultrahyperbolic signature which support higher rank Killing tensors or possess maximally superintegrable geodesic flows have been built along similar lines. Hidden symmetries of the Eisenhart lift metrics and the Dirac equation with flux have been studied in [13].…”

Hidden Symmetries of Ricci-Flat Spacetime and Integrable System With Harmonic Potentials

“…В частности, в недавней работе [11] были получены вакуумные решения уравнения Эйнштейна, допускающие тензоры Киллинга ранга 3 и 4, в пространствах ультрагиперболической сигнатуры (2,q) с q=2,3,4, а в [12] был построен новый класс (анти-)самодуальных Риччи--плоских многообразий ультрагиперболической сигнатуры, допускающих неприводимый тензор Киллинга второго ранга и обладающих максимально суперинтегрируемыми потоками геодезических.…”