Self-dual metrics with maximally superintegrable geodesic flows

TL;DR

This paper constructs specific self-dual, geodesically complete spacetimes with maximally superintegrable geodesic flows using Eisenhart lift, highlighting the role of second rank Killing tensors in ultrahyperbolic signatures.

Contribution

It introduces a new class of spacetimes with superintegrable geodesic flows characterized by second rank Killing tensors, extending to ultrahyperbolic signatures.

Findings

Construction of self-dual, geodesically complete spacetimes with superintegrable flows

Identification of second rank Killing tensors in these spacetimes

Extension to ultrahyperbolic signatures with similar properties

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.