NULL LEGENDRE CURVES IN LORENTZIAN HYPERSURFACES OF 5-DIMENSIONAL COSYMPLECTIC B-METRIC MANIFOLDS

Abstract

The object of study in the present paper is a class of null
curves in Lorentzian hypersurfaces $M$ of a 5-dimensional cosymplectic $B$-metric manifold \overline{M}, which are Legendre curves in the ambient manifold. We construct a basis along the examined curves through the almost
contact $B$-metric structure of  \overline{M} and the induced objects in $M$. By using this basis, we prove that there exists a unique Cartan frame for
the curves belonging to the investigated class. We show that if the
Lorentzian hypersurface $M$ is totally geodesic (resp. totally umbilical),
then the curve is geodesic (resp. non-geodesic). Special attention is
paid to the case when $M$ is totally umbilical. We obtain that if $M$ is an
extrinsic sphere, then the studied curves are helices. We construct an
example of a helix belonging to the considered class of null curves in a 4-
dimensional anti-de Sitter space $H^4_1$, which is a Lorentzian hypersurface
of $\mathbf{R}^5_2$, endowed with a cosymplectic $B$-metric structure.