Biharmonic curves in 3-dimensional Sasakian space forms

Abstract

We show that every proper biharmonic curve in a 3-dimensional Sasakian space form of constant holomorphic sectional curvature H is a helix (both of whose geodesic curvature and geodesic torsion are constants). In particular, if H ≠ 1, then it is a slant helix, that is, a helix which makes constant angle α with the Reeb vector field with the property \(\kappa^{2}+\tau^{2}=1+(H-1)\sin^{2}\alpha\). Moreover, we construct parametric equations of proper biharmonic herices in Bianchi–Cartan–Vranceanu model spaces of a Sasakian space form.