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

TL;DR

This paper studies non-null curves in strict Walker 3-manifolds, showing they lie in flat cylinders with null axes and exploring the geometric implications of totally geodesic cylinders.

Contribution

It characterizes the local geometry of non-null curves with zero torsion in strict Walker 3-manifolds and constructs explicit examples illustrating these properties.

Findings

Non-null curves lie in flat cylinders with null axes

Construction of explicit examples of such curves and cylinders

Implications of totally geodesic cylinders on ambient geometry

Abstract

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