Notes concerning Codazzi pairs on almost anti-Hermitian manifolds

Let $\mathrm{\nabla }$ be a linear connection on an $2n$-dimensional almost anti-Hermitian manifold $M$\ equipped with an almost complex structure $J$, a pseudo-Riemannian metric $g$ and the twin metric $G=g\circ J$. 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 relations of these conjugate connections, we prove a result stating that conjugations along with an identity operation together act as a Klein group. Secondly, we give some results exhibiting occurrences of Codazzi pairs which generalize parallelism relative to $\mathrm{\nabla }$. Under the assumption that $(\mathrm{\nabla },J)$ being a Codazzi pair,
we derive a necessary and sufficient condition the almost anti-Hermitian manifold $(M,J,g,G)$ is an anti-K"{a}hler relative to a torsion-free linear connection $\mathrm{\nabla }$. Finally, we investigate statistical structures on $M$ under $\mathrm{\nabla }$ ($\mathrm{\nabla }$ is a $J-$invariant torsion-free connection).