Geodesics of three-dimensional walker manifolds

Abstract

We review some basic facts of Lorentzian geometry including causality and geodesic completeness. We depict the properties of curves and planes in threedimensional Minkowski space. We deffne the Walker manifolds, that is, a Lorentzian manifold which admits a parallel degenerate distribution. We calculate the Christoffel symbols and Levi-Civita connection components, Riemann curvature and Ricci curvature components for an arbitrary three-dimensional Walker manifold and strictly Walker manifold. Finally, we derive the geodesic equations of a three-dimensional Walker manifold and investigate the geodesic curves in it, particularly the ones with a constant component. We prove that any straight line with a constant third component is a geodesic in any Walker manifold with the causality depending on its second component. We prove that the existence of a geodesic in a Walker manifold with a linear third component implies that the manifold is strict. We also show that any three-dimensional Walker manifold is geodesically complete.