Sharp inequalities and solitons for statistical submersions

TL;DR

This paper establishes sharp curvature inequalities and explores Ricci-Bourguignon solitons in the context of statistical submersions, linking geometric properties with soliton structures and deriving related equations.

Contribution

It introduces new sharp inequalities for statistical submersions and characterizes fibers as Ricci-Bourguignon solitons, advancing the understanding of geometric structures in statistical manifolds.

Findings

Proved sharp inequalities involving Ricci and scalar curvatures.

Characterized fibers of statistical submersions as Ricci-Bourguignon solitons.

Derived a Poisson equation for gradient-type vertical potential vector fields.

Abstract

In this research article, initially, we prove some sharp inequalities on statistical submersions involving Ricci and scalar curvatures of the statistical manifolds. In addition, we establish the geometrical bearing on statistical submersions in terms of Ricci-Bourguignon soliton. Moreover, we characterize the fibers of a statistical submersion as Ricci-Bourguignon solitons with conformal vector field. Finally, in the particular case when the vertical potential vector field of the Ricci-Bourguignon soliton is of gradient type, we derive a Poisson equation for a statistical submersion.