Some discussions on null helices in 3D semi-Riemannian manifold

TL;DR

This paper explores the geometric properties of null helices on totally umbilical submanifolds within 3D semi-Riemannian manifolds, revealing equations and invariants that characterize these curves in complex geometric settings.

Contribution

It provides new insights into null helices on totally umbilical submanifolds in semi-Riemannian manifolds, including equations and invariants specific to this context.

Findings

Derived equations characterizing null helices.

Identified invariants related to null Frenet frames.

Analyzed the impact of the totally umbilical condition.

Abstract

In this study, the geometric properties of null helices on a totally umbilical submanifold within a three-dimensional semi-Riemannian manifold are investigated. The pseudo-Riemannian metric structure of semi-Riemannian manifolds 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 characterising null helices are obtained. This study aims to contribute to the theory of null curves on semi-Riemannian manifolds and the geometry of special submanifolds.