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

Abstract

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.

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

“…Following the approach of [5] (or [6]) many researchers have investigated the geometry of hull submanifolds which include, but not limited to, [1], [2], [4], [9], [9], [11], [12], [13], [14] and [19]. In a recent paper [13], the author initiated the study of ascreen null hypersurfaces of Lorentzian concircular structure (LCS)manifolds, in which it was discovered that such hypersurfaces admits a symmetric Ricci tensor. The rest of the paper is arranged as follows; Section 2 focusses on the basic notions of null hypersurfaces and Lorentzian concircular structure (LCS)manifolds needed in the rest of the paper.…”

“…where a and b are smooth functions, given by a = θ(N ) and b = θ(ξ). It was proved in [13] that ζ is never tangent or transversal to M . Furthermore, it was shown, in the same paper, that φT M ⊥ and φtr(T M ) can not be considered as subbundles of S(T M ) as it is often done in the Sasakian case (see [6] for detailes).…”

“…As B is symmetric, it is easy to see from (3.5) that C is also symmetric. Using the functions a and b, it was shown in [13] that the Ricci tensor of an ascreen null hypersurface of a (LCS)-manifold is actually symmetric. In the following result, we supply a different proof to that in [13].…”

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

“…Following the approach of [5] (or [6]) many researchers have investigated the geometry of hull submanifolds which include, but not limited to, [1], [2], [4], [9], [9], [11], [12], [13], [14] and [19]. In a recent paper [13], the author initiated the study of ascreen null hypersurfaces of Lorentzian concircular structure (LCS)manifolds, in which it was discovered that such hypersurfaces admits a symmetric Ricci tensor. The rest of the paper is arranged as follows; Section 2 focusses on the basic notions of null hypersurfaces and Lorentzian concircular structure (LCS)manifolds needed in the rest of the paper.…”

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

“…More precisely, φT M ⊥ ∕ ⊂ S(T M ) and φtr(T M ) ∕ ⊂ S(T M ). Moreover, φT M ⊥ ∩ T M ⊥ = {0} and φtr(T M ) ∩ tr(T M ) = {0} (see[12] for more details).In view of[8], we will say that a null hypersurface (M, g) of a (LCS)-manifold (M , g) is ascreen if the characteristic vector field ζ belongs to S(T M ) ⊥ (= T M ⊥ ⊕ tr(T M )). Equivalently, M is ascreen if W = 0.…”