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

TL;DR

The paper studies helix curves on the 3-dimensional unit tangent bundle $T_1M^2(\kappa)$ of a pseudo-Riemannian surface of constant curvature, endowed with a pseudo-Riemannian $g$-natural metric of Kaluza–Klein type. It shows that helices directed by the geodesic flow are circular in the sense that their curvature and (where applicable) torsion are constant; vertical helices are always Legendre, while non-vertical ones occur only when the base curve is a geodesic with a parallel transport field along it, under KK constraints. Non-Kaluza–Klein metrics admit non-degenerate Frenet helices with constant curvature and torsion, and Cartan null helices are characterized via Cartan frames. The results provide a complete framework for horizontal and oblique helices, including explicit differential relations for the base curve and accompanying vector field and highlighting the role of the metric type in the existence and nature of helices on $T_1M^2(\kappa)$. Overall, the work integrates contact/paracontact geometry, $g$-natural metrics, and Frenet-Cartan formalisms to classify helix curves in this pseudo-Riemannian setting.

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.

Figures (2)

• Figure 1: Example of geodesic helices in $T_1\mathbb{S}^2_1$
• Figure 2: Example of an oblique helix curve in $T_1\mathbb{S}^2$.

Theorems & Definitions (25)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Proposition 1.1
• Proposition 1.2
• Proposition 1.3