Statistical connections on decomposable Riemann manifold

Let (M, g, phi) be an n-dimensional locally decomposable Riemann manifold, that is, g(phi X, Y) = g(X, phi Y) and del phi = 0, where del 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, phi)-connection. A statistical alpha-connection is a torsion-free connection such that (del) over barg = alpha C, where C is a completely symmetric (0, 3)-type cubic form. The aim of this article is to use connection (del) over bar and product structure phi in the same equation, which is possible by writing the cubic form C in terms of the product structure phi. We examine some curvature properties of the new connection and give examples of it.