Notes concerning Kähler and anti-Kähler structures on quasi-statistical manifolds

Abstract

Let $(\acute{N},g,\nabla )$\ be a $2n$-dimensional quasi-statistical manifold that admits a pseudo-Riemannian metric $g$ (or $h)$ and a linear connection $\nabla$ with torsion. This paper aims to study an almost Hermitian structure $(g,L)$ and an almost anti-Hermitian structure $(h,L)$ on a quasi-statistical manifold that admit an almost complex structure $L$. Firstly, under certain conditions, we present the integrability of the almost complex structure $L$. We show that when $d^{\nabla }L=0$ and the condition of torsion-compatibility are satisfied, $(\acute{N},g,\nabla ,$ $L)$ turns into a Kähler manifold. Secondly, we give necessary and sufficient conditions under which $(\acute{N},h,\nabla ,L)$ is an anti-Kä% hler manifold, where $h$ is an anti-Hermitian metric. Moreover, we search the necessary conditions for $(\acute{N},h,\nabla ,L)$ to be a quasi-Kähler-Norden manifold.