On the invariant trajectories of lightlike paths near central potential singularities in semi-Riemannian manifolds of index 2

Abstract

In this study, the geometric properties of null helices on a totally umbilical submanifold within a three-dimensional semi-Riemannian manifold of index 2 are investigated. The pseudo-Riemannian metric structure of semi-Riemannian manifold of index 2 and the fact that the submanifold is totally umbilical complicate the differential geometry of null helices. The study uses the given null Frenet frame to reveal the local properties of null helices (the curvatures $h,k_1,k_2$ of the null curves). By considering the degenerate metric condition of null helices due to the null tangent vector and the structure of the totally umbilical submanifold, equations and invariants characterizing null helices are obtained. Furthermore, this study establishes a novel structural rigidity theorem demonstrating that the simultaneous preservation of these degenerate curvature invariants under higher-order ambient parallel transport algebraically forces the collapsing of the mean curvature vector field ($H\equiv 0$), thereby identifying a clean boundary distinguishing minimal and totally geodesic embeddings in pseudo-Euclidean target spaces.