Two-jets of conformal fields along their zero sets

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.