Biharmonic curves in $3$-dimensional Lorentzian Sasakian space forms
Commun. Korean Math. Soc. 2020 Vol. 35, No. 3, 967-977
https://doi.org/10.4134/CKMS.c190393
Published online July 5, 2020
Ji-Eun Lee
Chonnam National University
Abstract : 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)
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