Zeros of conformal fields in any metric signature

TL;DR

This paper generalizes the understanding of zero sets of conformal vector fields to pseudo-Riemannian manifolds of any signature, revealing their structure as totally umbilical conifold varieties with possible quadric singularities.

Contribution

It extends previous Riemannian results to indefinite metrics, characterizing the zero sets of conformal fields in all metric signatures including Lorentzian.

Findings

Zero sets are totally umbilical conifold varieties.

Singularities occur only in indefinite metrics.

Generalization of Riemannian results to pseudo-Riemannian manifolds.

Abstract

The connected components of the zero set of any conformal vector field, in a pseudo-Riemannian manifold of arbitrary signature, are shown to be totally umbilical conifold varieties, that is, smooth submanifolds except possibly for some quadric singularities. The singularities occur only when the metric is indefinite, including the Lorentzian case. This generalizes an analogous result in the Riemannian case, due to Belgun, Moroianu and Ornea (2010).