Contact Whirl Curves in Sasakian Lorentzian 3-Manifolds

Abstract

We introduce and study \emph{contact whirl curves} in three-dimensional Lorentzian contact manifolds, with emphasis on the Sasakian setting. This notion refines the concept of whirl curves by encoding the interaction between the adapted frame of a curve and the ambient contact structure through the Reeb vector field. For non-geodesic unit-speed contact whirl curves, we derive a differential equation governing the torsion in terms of the Frenet invariants and the contact data. In the Lorentzian Sasakian setting, this leads to rigidity phenomena of Lancret type. In particular, we prove that every non-geodesic Legendre Frenet curve is automatically a contact whirl curve, and consequently has constant torsion $τ=1$. We also investigate the interaction between contact whirl curves and magnetic trajectories associated with the canonical contact magnetic field. We show that every non-geodesic curve which is simultaneously magnetic and contact whirl must be Legendre, and we obtain an explicit expression for its torsion in terms of the tensor $h=\frac12\mathcal L_ξΦ$. In the Sasakian case, this reduces to the universal law $τ=1$. Finally, in the Lorentzian Heisenberg group endowed with its standard Sasakian structure, we derive a coordinate form of the whirl condition and use it to produce explicit examples, including a construction by quadratures of non-Legendre contact whirl curves and a horizontal helicoidal Legendre family.