Chen-like inequalities on null hypersurfaces with closed rigging of a Lorentzian manifold

We study null hypersurfaces of a Lorentzian manifold with a closed rigging for the hypersurface. We derive inequalities involving Ricci tensors, scalar curvature, squared mean curvatures for a null hypersurface with a closed rigging of a Lorentzian space form and for a screen homothetic null hypersurface of a Lorentzian manifold. We also establish a generalized Chen–Ricci inequality for a screen homothetic null hypersurface of a Lorentzian manifold with a closed rigging for the hypersurface.

A characterization of minimal ascreen null hypersurfaces of $(LCS)$-space forms

In the present paper, we study nontotally geodesic minimal ascreen null hypersurface, $M$, of a Lorentzian concircular structure $(LCS)$-space form of constant curvature $0$ or $1$. We prove that; if the Ricci tensor of $M$ is parallel with respect to any leaf of its screen distribution, then $M$ is isometric to a product of a null curve and spheres.

On Einstein null hypersurfaces of $(LCS)_{n+2}$-space forms

We show that ascreen null hypersurfaces of an $(n+2)$-dimensional Lorentzian concircular structure $(LCS)_{n+2}$-manifold admits an induced Ricci tensor. We, therefore, prove, under some geometric conditions, that an Einstein ascreen null hypersurface is locally a product of null curves and products of spheres.

Geometry of isoparametric null hypersurfaces of Lorentzian manifolds

We define two types of null hypersurfaces as; isoparametric and quasi isoparametric null hypersurfaces of Lorentzian space forms, based on the two shape operators associated with a null hypersurface. We prove that; on any screen conformal isoparametric null hypersurface, the screen geodesics lie on circles in the ambient space. Furthermore, we prove that the screen distributions of isoparametric (or quasi-parametric) null hypersurfaces with at most two principal curvatures are generally Riemannian products. Several examples are also given to illustrate the main concepts.

Some results on null hypersurfaces in $(LCS)$-manifolds

We prove that a Lorentzian concircular structure $ (LCS)$-manifold does not admit any null hypersurface which is tangential or transversal to its characteristic vector field. Due to the above, we focus on its ascreen null hypersurfaces, and show that such hypersurfaces admit a symmetric Ricci tensor. Furthermore, we prove that there is no any totally geodesic ascreen null hypersurfaces of a conformally flat $(LCS)$-manifold.