AI 中文总结
本文从直接里奇曲率视角，结合统计浸没的里奇曲率关系与Hineva代数不等式，首次得到统计浸没的希内瓦型下里奇曲率估计，为统计浸没提供统一里奇曲率方法并获得双侧里奇曲率界。
AI 中文摘要
本文从直接里奇曲率视角研究统计浸没的里奇曲率界。尽管已建立统计浸没的陈-里奇不等式，但仍需补充下界以获得里奇曲率的双侧描述。受此启发，我们结合统计浸没的里奇曲率关系与希内瓦（Hineva）代数不等式。首先，我们沿垂直分布得到陈-里奇上界和希内瓦型下界，其表达式涉及纤维的固有里奇曲率及基本张量T和T*。这两个估计共同为垂直里奇曲率提供上下界。随后，我们推导沿水平分布涉及基本张量A和A*的陈-里奇不等式。通过结合垂直与水平曲率关系，我们进一步得到混合分布对应的陈-里奇和希内瓦型估计。等式条件由基本张量及其对偶的分量表征，本文给出具体实例说明等式与严格不等式情形。因此，本文提供了一种统计浸没的统一里奇曲率方法，尤其首次给出统计浸没的希内瓦型下里奇曲率估计，进而得到双侧里奇曲率界。
英文摘要
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.
