Communications of the
Korean Mathematical Society CKMS

In this article, we find the necessary and sufficient condition for a proper biharmonic Frenet curve in the Lorentzian Sasakian space forms $\mathcal{M}_1^3 (H)$ except the case constant curvature $-1$. Next, we find that for a slant curve in a $3$-dimensional Sasakian Lorentzian manifold, its ratio of ``geodesic curvature" and ``geodesic torsion $-1$" is a constant. We show that a proper biharmonic Frenet curve is a slant pseudo-helix with $\kappa^2-\tau^2=-1+\varepsilon_1(H+1)\eta(B)^2$ in the Lorentzian Sasakian space forms $\mathcal{M}_1^3 (H)$ except the case constant curvature $-1$. As example, we classify proper biharmonic Frenet curves in $3$-dimensional Lorentzian Heisenberg space, that is a slant pseudo-helix.

Keywords: Slant curves, Legendre curves, biharmonic, Lorentzian Sasakian space forms

MSC numbers: Primary 53B25, 53C25

Supported by: The author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2019R1l1A1A01043457)