Notes concerning Codazzi pairs on almost anti-Hermitian manifolds

Let ∇ be a linear connection on a 2n-dimensional almost anti-Hermitian manifold M equipped with an almost complex structure J, a pseudo-Riemannian metric g and the twin metric G = g ◦ J. In this paper, we first introduce three types of conjugate connections of linear connections relative to g, G and J. We obtain a simple relation among curvature tensors of these conjugate connections. To clarify the relations of these conjugate connections, we prove a result stating that conjugations along with an identity operation together act as a Klein group, which is analogue to the known result for the Hermitian case in [2]. Secondly, we give some results exhibiting occurrences of Codazzi pairs which generalize parallelism relative to ∇. Under the assumption that (∇, J) being a Codazzi pair, we derive a necessary and sufficient condition the almost anti-Hermitian manifold (M, J, g, G) is an anti-Kähler relative to a torsion-free linear connection ∇. Finally, we investigate statistical structures on M under ∇ (∇ is a J–parallel torsion-free connection).