Totally umbilical surfaces in three-manifolds with a parallel null vector field

TL;DR

This paper classifies totally umbilical surfaces in Walker three-manifolds with a parallel null vector field, showing they are either totally geodesic or the ambient space is locally conformally flat.

Contribution

It extends the classification of totally umbilical surfaces to a special class of pseudo-Riemannian three-manifolds, using techniques from homogeneous Riemannian geometry.

Findings

Such surfaces are either totally geodesic or the ambient manifold is locally conformally flat.

The classification aligns with known results in homogeneous Riemannian three-manifolds.

The proof employs a key technique from recent classifications in Riemannian geometry.

Abstract

We study non-degenerate, totally umbilical surfaces of a special class of pseudo-Riemannian manifolds, namely Walker three-manifolds. We show that such surfaces are either one of a totally geodesic family described by Calvaruso and Van der Veken or the ambient manifold must be locally conformally flat (here the surface can also be totally geodesic). The proof makes use of a key technique deployed by Manzano and Soaum in their recent classification of totally umbilical surfaces in homogeneous Riemannian three-manifolds.