Totally umbilical radical transversal lightlike hypersurfaces of Kähler-Norden manifolds of constant totally real sectional curvatures

Abstract

In this paper we study curvature properties of semi-symmetric type of totally umbilical radical transversal lightlike hypersurfaces $(M,g)$ and $(M,\tilde{g})$ of a Kähler-Norden manifold $(\overline{M},\overline{J},\overline{g},\overline{\tilde{g}})$ of constant totally real sectional curvatures $\overline{\nu }$ and $\overline{\tilde{\nu }}$ ($g$ and $\tilde{g}$ are the induced metrics on $M$ by the Norden metrics $\overline{g}$ and $\overline{\tilde{g}}$,respectively). We obtain a condition for $\overline{\tilde{\nu }}$ (resp. $\overline{\nu }$) which is equivalent to each of the following conditions: $(M,g)$ $(resp.(M,\tilde{g}))$ is locally symmetric, semi-symmetric, Ricci semi-symmetric and almost Einstein. We construct an example of a totally umbilical radical transversal lightlike hypersurface, which is locally symmetric, semi-symmetric, Ricci semi-symmetric and almost Einstein.