Helix curves of the unit tangent bundle of a pseudo-Riemannian surface

TL;DR

This paper classifies helix curves on the unit tangent bundle of a pseudo-Riemannian surface with constant curvature, showing they are all circular helices with constant curvature and torsion.

Contribution

It provides a complete classification of helix curves on the unit tangent bundle of pseudo-Riemannian surfaces, revealing their circular nature with constant curvature and torsion.

Findings

Helix curves are classified as spacelike, timelike, or null.

All helix curves are circular helices with constant curvature and torsion.

The classification applies to surfaces with constant Gaussian curvature and specific metrics.

Abstract

In this paper, we classify helix (spacelike, timelike and null) curves, directed by the geodesic flow vector field, on the (3-dimensional) unit tangent bundle of a pseudo-Riemannian surface of constant Gaussian curvature endowed with a pseudo-Riemannian $g$-natural metric of Kaluza-Klein type. We find, in particular, that every such helix curve is, in fact, a circular helix in the senses that its curvature and torsion are constant.