Surfaces in a strict Walker 3-manifold that contain non-null curves with zero torsion

Abstract

Given a non-null curve ${\hat{I}}^3$ in a strict Walker 3-manifold, first we show that (locally) ${\hat{I}}^3$ lies in a flat cylinder with a null axis. Secondly, we construct an example of such a curve ${\hat{I}}^3$ and such a cylinder $S$ that contains ${\hat{I}}^3$ . In particular, the hypothesis that $S$ is totally geodesic has some consequence on the geometry of the ambient Walker 3-manifold.