Slant and Legendre Null Curves in 3-Dimensional Sasaki-like Almost Contact B-Metric Manifolds

TL;DR

This paper studies special null curves called slant and Legendre in 3D Sasaki-like almost contact B-metric manifolds, deriving conditions for their curvatures and characterizations as helices or null cubics.

Contribution

It provides explicit formulas for the Frenet frames and curvatures of these null curves, establishing when their curvatures are constant and characterizing their geometric types.

Findings

Curvatures are constant iff a specific function on the manifold is constant.

Slant null curves are generalized helices under certain conditions.

Legendre null curves can be null cubics with specific properties.

Abstract

Object of study in the present paper are slant and Legendre null curves in 3-dimensional Sasaki-like almost contact B-metric manifolds. For the examined curves we express the general Frenet frame and the Frenet frame for which the original parameter is distinguished, as well as the corresponding curvatures, in terms of the structure on the manifold. We prove that the curvatures of a framed null slant and Legendre curve are constants if and only if a specific function for the considered manifolds is a constant. We find a necessary and sufficient condition a slant null curve to be a generalized helix and a Legendre null curve to be a null cubic. For some of investigated curves we show that they are non-null slant or Legendre curves with respect to the associated B-metric on the manifold. We give examples of the examined curves. Some of them are constructed in a 3-dimensional Lie group as Sasaki-like manifold and their matrix representation is obtained.