Submanifold Theory in Holomorphic Statistical Manifolds

Abstract

A statistical manifold is a smooth manifold equipped with a pair of a Riemannian metric and a torsion-free affine connection satisfying the Codazzi equation. We naturally have various dualistic geometric objects on it. In this article, the basics for statistical submanifolds in holomorphic statistical manifolds are given. We define the sectional curvature for a statistical structure, and study CR-submanifolds in a holomorphic statistical manifold of constant holomorphic sectional curvature. We prove that this sectional curvature of such a space vanishes if it admits a totally umbilical and a dual-totally umbilical generic submanifolds. Furthermore, we show that a Lagrangian submanifold is of constant sectional curvature if the statistical shape operator and its dual operator commute. Similarly, we generalize several theorems in the classical CR-submanifold theory.