Smarandache curves and their properties on null curves in lightlike cone space $\mathbb{Q}_{2}^{3}$

TL;DR

This paper explores the geometric properties of Smarandache curves derived from null curves in lightlike cone space, revealing how their invariants are influenced by the space's indefinite metric and providing insights into degenerate structures in differential geometry.

Contribution

It introduces a detailed analysis of Smarandache curves on null curves within lightlike cone space, highlighting the effects of the indefinite metric on their geometric invariants.

Findings

Null curves exhibit richer geometry in lightlike cone space.

Smarandache curves inherit properties from original null curves.

The space's metric significantly influences the characteristics of Smarandache curves.

Abstract

This study investigates the differential geometric properties of Smarandache curves derived from null curves defined in the ligtlike cone space $% Q_{3}^{2}\subset E_{2}^{4}$. The indefinite metric structure causes the null vectors, and hence the null curves, to have a richer geometry in this space than in Euclidean or Minkowski spaces. In this study, we analyse the kinematic properties of the null curve using the null natural Frenet frame $% \{x,\xi ,N,W\}$. We then investigate the bending, torsion, and other geometric invariants of Smarandache curves constructed as linear combinations of these frame vectors (i.e., combinations of tangent, normal, or binormal vectors). The findings reveal how the original properties of the null curve are transferred to the Smarandache curves and how the metric of this particular space affects the characteristics of the Smarandache curves. This analysis provides a new perspective on the relationships between constrained and degenerate structures in differential geometry and light-like particle dynamics in theoretical physics.