Geometric Inequalities for $CR$ $\delta$-Invariant on Generic Statistical Submanifolds

Abstract

B.-Y. Chen \cite{chen} introduced the notion of $CR$ $\delta$-invariant on $CR$-submanifolds. Recently, F. Al-Solamy et al. \cite{s1,s2} and I. Mihai et al. \cite{ion}, respectively, established optimal inequalities for this invariant on anti-holomorphic submanifolds in complex space forms and for generic submanifolds in Sasakian space forms. In the present paper, we obtain two optimal inequalities involving the $CR$ $\delta$-invariant for a generic statistical submanifold in a holomorphic statistical manifold of constant holomorphic sectional curvature. Finally, we consider a generic statistical submersion from a holomorphic statistical manifold onto a statistical manifold.