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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.
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참고문헌 (14)
참고문헌 신청
D. E. Blair / 2002 / Riemannian geometry of contact and symplectic manifolds / Birkh¨auser Boston, Inc.
C. Calin / 2012 / Slant curves in 3-dimensional fKenmotsu manifolds / J. Math. Anal. Appl 394(1):400~407
G. Calvaruso / 2011 / Contact Lorentzian manifolds / Differential Geom. Appl. 29:S41~S51
G. Calvaruso / 2010 / Contact pseudo-metric manifolds / Differential Geom. Appl 28(5):615~634
B. -Y. Chen / 1991 / Biharmonic surfaces in pseudo-Euclidean spaces / Mem. Fac. Sci. Kyushu Univ. Ser. A 45(2):323~347
C. Calin / 2012 / Slant curves in 3-dimensional fKenmotsu manifolds / J. Math. Anal. Appl 394(1):400~407
G. Calvaruso / 2011 / Contact Lorentzian manifolds / Differential Geom. Appl. 29:S41~S51
G. Calvaruso / 2010 / Contact pseudo-metric manifolds / Differential Geom. Appl 28(5):615~634
B. -Y. Chen / 1991 / Biharmonic surfaces in pseudo-Euclidean spaces / Mem. Fac. Sci. Kyushu Univ. Ser. A 45(2):323~347
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