Essential points of conformal vector fields

TL;DR

This paper characterizes essential points of conformal vector fields on Riemannian manifolds, showing they are isolated zeros, and demonstrates that zero sets are totally umbilical, providing geometric insights into conformal symmetries.

Contribution

It establishes that essential points are precisely the isolated zeros of conformal vector fields and describes the geometric structure of their zero sets.

Findings

Essential points are exactly the isolated zeros of conformal vector fields.

Zero sets of conformal vector fields are totally umbilical.

Provides a geometric characterization of conformal vector fields' zeros.

Abstract

For a conformal vector field $\xi$ on a Riemannian manifold, we say that a point is essential if there is no local metric in the conformal class for which $\xi$ is Killing. We show that the only essential points are isolated zeros of $\xi$. As an application, we show that every connected component of the zero set of $\xi$ is totally umbilical.