Ricci curvature bounds for Statistical Submersions
In this paper, we study Ricci curvature bounds for statistical submersions from a direct Ricci-curvature perspective. Although Chen--Ricci inequalities for statistical submersions have already been established, a complementary lower estimate is needed to obtain a two-sided description of the Ricci curvature. Motivated by this observation, we combine the Ricci curvature relations of a statistical submersion with Hineva's algebraic inequality. We first obtain a Chen--Ricci upper estimate and a Hineva-type lower estimate along the vertical distribution, expressed in terms of the intrinsic Ricci curvature of the fibres and the fundamental tensors $T$ and $T^{*}$. The two estimates together provide upper and lower bounds for the vertical Ricci curvature. We then derive a Chen--Ricci inequality along the horizontal distribution involving the fundamental tensors $A$ and $A^{*}$. By combining the vertical and horizontal curvature relations, we further obtain corresponding Chen--Ricci and Hineva-type estimates for the mixed distribution. The equality conditions are characterized in terms of the components of the fundamental tensors and their duals. Explicit examples are given to illustrate both equality and strict inequality cases. Thus, the paper provides a unified Ricci-curvature approach to statistical submersions and, in particular, gives the first Hineva-type lower Ricci curvature estimates for statistical submersions, leading to two-sided Ricci curvature bounds.