On lightlike hypersurfaces of a GRW space-time (Q2902043)

A generalized Robertson-Walker space-time (GRW) is defined as a warped product \(L_{1}^{n+1}(c,f)=(I \times_{f} F,\bar{g})\), where \(I\subset \mathbb{R}\) is an open interval, \((F,g_{c})\) is an \(n\)-dimensional Riemannian manifold of constant sectional curvature \(c\) and \(\bar{g}=-dt^{2}+f^{2}(t)g_{c}\). The author investigates light-like hypersurfaces of a GRW space-time under several different conditions.NEWLINENEWLINESuppose \((M,g)\) is a light-like hypersurface of \(L_{1}^{n+1}(c,f)\). Several results are obtained as below. (a) \(M\) is curvature invariant if and only if \(L_{1}^{n+1}(c,f)\) is of constant curvature. (b) If the second fundamental form \(h\) is parallel, then \(L_{1}^{n+1}(c,f)\) has constant sectional curvature. (c) If \(M\) is totally umbilical or the screen distribution \(S(TM)\) with \(rank>1\) is totally umbilical, a partial differential equation is derived. Also, this paper investigates another equivalent condition for the given constant sectional curvature \(L_{1}^{n+1}(c,f)\) given in terms of null sectional curvatures and null Ricci curvatures.NEWLINENEWLINEIn the past, the author also provided other results on light-like hypersurfaces of indefinite cosymplectic manifolds and indefinite Sasakian manifolds in [\textit{T. H. Kang} and \textit{S. K. Kim}, Int. Math. Forum 2, No. 65--68, 3303--3316 (2007; Zbl 1151.53016); \textit{T. H. Kang} et al., Indian J. Pure Appl. Math. 34, No. 9, 1369--1380 (2003; Zbl 1046.53053)].