Interaction of Codazzi Pairs with Almost Para Norden Manifolds

Abstract

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