Generalized Taub-NUT metrics and Killing-Yano tensors

TL;DR

This paper investigates the relationship between Stäckel-Killing tensors and Killing-Yano tensors in Riemannian manifolds, focusing on generalized Euclidean Taub-NUT metrics and their symmetries, revealing limitations in expressing certain tensors as products.

Contribution

It re-derives a necessary condition for Stäckel-Killing tensors to be contracted products of Killing-Yano tensors and applies it to generalized Taub-NUT metrics, highlighting unique properties of the original Taub-NUT case.

Findings

Most generalized Taub-NUT metrics do not allow expressing the Runge-Lenz vector's tensors as Killing-Yano products.

The original Taub-NUT metric uniquely admits such a tensor product representation.

The study clarifies the special symmetry properties of the original Taub-NUT space.

Abstract

A necessary condition that a St\"ackel-Killing tensor of valence 2 be the contracted product of a Killing-Yano tensor of valence 2 with itself is re-derived for a Riemannian manifold. This condition is applied to the generalized Euclidean Taub-NUT metrics which admit a Kepler type symmetry. It is shown that in general the St\"ackel-Killing tensors involved in the Runge-Lenz vector cannot be expressed as a product of Killing-Yano tensors. The only exception is the original Taub-NUT metric.