Killing tensors and a new geometric duality

TL;DR

This paper establishes a duality between geometries admitting Killing tensors and those with metrics defined by them, extending to spinning spaces with torsion and revealing new supersymmetries, with applications to black hole and Taub-NUT metrics.

Contribution

It introduces a novel geometric duality relating Killing tensors to metrics, including extensions to spinning spaces with torsion and supersymmetry implications.

Findings

Dual relation between Killing tensor geometry and metric geometry

Extension to spinning spaces with torsion and supersymmetry

Application to Kerr-Newman and Taub-NUT metrics

Abstract

We present a theorem describing a dual relation between the local geometry of a space admitting a symmetric second-rank Killing tensor, and the local geometry of a space with a metric specified by this Killing tensor. The relation can be generalized to spinning spaces, but only at the expense of introducing torsion. This introduces new supersymmetries in their geometry. Interesting examples in four dimensions include the Kerr-Newman metric of spinning black-holes and self-dual Taub-NUT.