Biharmonic curves in $3$-dimensional Lorentzian Sasakian space forms
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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.
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.
참고문헌 (14)
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