Quasi-Statistical Manifolds with Almost Hermitian and Almost Anti-Hermitian Structures

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