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

Abstract Let ( M, g, ∇) be a 2 n -dimensional quasi-statistical manifold that admits a pseudo-Riemannian metric g (or h ) and a linear connection ∇ 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 ∇ J = 0 and the condition of torsion-compatibility are satisfied, ( M, g, ∇, J ) turns into a Kähler manifold. Secondly, we give necessary and sufficient conditions under which ( M, h, ∇, J ) is an anti-Kähler manifold, where h is an anti-Hermitian metric. Moreover, we search the necessary conditions for ( M, h, ∇, J ) to be a quasi-Kähler-Norden manifold.