Totally umbilical radical transversal lightlike hypersurfaces of K\"ahler-Norden manifolds of constant totally real sectional curvatures

TL;DR

This paper investigates the curvature properties of specific lightlike hypersurfaces in K"ahler-Norden manifolds with constant totally real sectional curvatures, establishing conditions for various geometric properties and providing an example.

Contribution

It introduces new curvature conditions for totally umbilical radical transversal lightlike hypersurfaces in K"ahler-Norden manifolds and constructs an explicit example demonstrating these properties.

Findings

Conditions for local symmetry, semi-symmetry, Ricci semi-symmetry, and almost Einstein properties.

Equivalence of curvature conditions with specific geometric properties.

Explicit example of a hypersurface with these properties.

Abstract

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