Statistical connections on decomposable Riemann manifold

Let $(M, g, \varphi)$ be an $n$-dimensional locally decomposable Riemann manifold, that is, $g(\varphi X, Y) = g(X, \varphi Y)$ and $\nabla \varphi = 0$, where $\nabla $ is Riemann (Levi-Civita) connection of metric $g$. In this paper, we construct a new connection on locally decomposable Riemann manifold, whose name is statistical ($\alpha, \varphi)$-connection. A statistical $\alpha $-connection is a torsion-free connection such that $% \overline{\nabla }g = \alpha C$, where $C$ is a completely symmetric $(0, 3)$% -type cubic form. The aim of this article is to use connection $\overline{% \nabla }$ and product structure $\varphi $ in the same equation, which is possible by writing the cubic form $C$ in terms of the product structure $% \varphi $. We examine some curvature properties of the new connection and give examples of it.