Interaction of Codazzi Pairs with Almost Para Norden Manifolds

In this paper, we research some properties of Codazzi pairs on almost para Norden manifolds. Let $(M_{2n},\ \varphi ,\ g,G)$ be an manifold. Firstly, $g$-conjugate connection, $G$-conjugate connection and $\varphi $-conjugate a linear $\mathrm{\nabla }$ $M_{2n}$ denoted by ${\mathrm{\nabla }}^{*\
 },\ {\mathrm{\nabla }}^{\dagger \ }}^{\varphi are defined it is demonstrated that the spaces connections, $\left(id,\ *,\dagger ,\varphi \right)$ acts as four-element Klein group. We also searched these three types conjugate
 connections. Then, $\left(\mathrm{\nabla },\varphi \right)\ ,\left(\mathrm{\nabla },g\right)$ },G\right)$ introduced them given. $R\ R^{*\ }$and $R^{\dagger }$are $(0,4)$-curvature tensors conjugate connections
 }\mathrm{\ }{\mathrm{\nabla }}^{*\ }$, respectively. The relationship among curvature investigated. condition $N_{\varphi }=0$ obtained, where Nijenhuis tensor field known
 integrability complex structure $ }=0$. addition, Tachibana operator applied to pure metric $g$ necessary sufficient $\left(M,\varphi g,G\right)$ being Kahler manifold found. Finally, examine $-invariant connections statistical