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

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.