NULL CARTAN CURVES IN STRICT WALKER THREE-MANIFOLDS

Source abstract

In this paper, we study null Cartan curves in strict Walker three-manifolds on intervals where theirtangent is not proportional to the distinguished parallel null vector field. On suchintervals, a null curve is determined locally by its transverse coordinate. An adaptednull frame yields a scalar function $\varphi=y''-\frac12 f_y$ that governs theacceleration and the Cartan frame. We determine the pseudo-arc parameter and Cartancurvature explicitly in terms of $\varphi$ and characterize the curves with vanishingCartan curvature. A local correspondence with null Cartan curves in Minkowskithree-space preserves the pseudo-arc parameter and curvature. We also construct curveswith any prescribed smooth Cartan curvature through an arbitrary point and describethe resulting equations for plane-wave profiles. Examples illustrate these results inflat and non-flat strict Walker metrics.