Üç boyutlu Walker manifoldlarda jeodezikler

We review some basic facts of Lorentzian geometry including causality and geodesic completeness. We depict the properties of curves and planes in three-dimensional Minkowski space. We define 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.