Notes concerning K\"ahler and anti-K\"ahler structures on quasi-statistical manifolds

TL;DR

This paper investigates conditions under which quasi-statistical manifolds with certain structures become Kähler, anti-Kähler, or quasi-Kähler-Norden, expanding the understanding of complex and anti-complex geometric structures in these manifolds.

Contribution

It provides new criteria for integrability and geometric structures on quasi-statistical manifolds, including conditions for Kähler, anti-Kähler, and quasi-Kähler-Norden structures.

Findings

Conditions for integrability of almost complex structures.

Criteria for Kähler and anti-Kähler structures on quasi-statistical manifolds.

Necessary conditions for quasi-Kähler-Norden structures.

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 K\"{a}hler manifold. Secondly, we give necessary and sufficient conditions under which $(\acute{N},h,\nabla ,L)$ is an anti-K\"{a}% 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-K\"{a}hler-Norden manifold.