Two-jets of conformal fields along their zero sets

TL;DR

This paper classifies the zero sets of conformal vector fields on pseudo-Riemannian manifolds into essential and nonessential types, analyzing their geometric properties and associated structures like null totally geodesic submanifolds and projective Killing forms.

Contribution

It introduces a detailed classification of zero sets of conformal fields, describing their geometric nature and associated invariants, which advances understanding of conformal geometry in arbitrary signatures.

Findings

Zero sets are of two types: essential and nonessential.

Essential components are null totally geodesic submanifolds with specific properties.

The 2-jet of the conformal field is locally constant along certain submanifolds.

Abstract

The connected components of the zero set of any conformal vector field $v$, in a pseudo-Riemannian manifold $(M,g)$ of arbitrary signature, are of two types, which may be called `essential' and `nonessential'. The former consist of points at which $v$ is essential, that is, cannot be turned into a Killing field by a local conformal change of the metric. In a component of the latter type, points at which $v$ is nonessential form a relatively-open dense subset that is at the same time a totally umbilical submanifold of $(M,g)$. An essential component is always a null totally geodesic submanifold of $(M,g)$, and so is the set of those points in a nonessential component at which $v$ is essential (unless this set, consisting precisely of all the singular points of the component, is empty). Both kinds of null totally geodesic submanifolds arising here carry a 1-form, defined up to multiplications by functions without zeros, which satisfies a projective version of the Killing equation. The conformal-equivalence type of the 2-jet of $v$ is locally constant along the nonessential submanifold of a nonessential component, and along an essential component on which the distinguished 1-form is nonzero. The characteristic polynomial of the 1-jet of $v$ is always locally constant along the zero set.