Hitoshi Furuhata

Submanifold Theory in Holomorphic Statistical Manifolds

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.

A Cylinder Theorem for Isometric Pluriharmonic Immersions

We prove a cylinder theorem for isometric pluriharmonic immersions of complete Kähler manifolds into semi-Euclidean spaces under an assumption concerning the index of relative nullity.

Holomorphic centroaffine immersions and the Lelieuvre correspondence

The rigidity and intrinsic characterization of holomorphic centroaffine immersions are given. We also obtain a method to construct nondegenerate holomorphic affine hypersurfaces from centroaffine immersions and metrics satisfying some conditions. Mathematics subject classification: 53A15.

Sasakian Statistical Manifolds II.

This article is a digest of [2, 3] with additional remarks on invariant submanifolds of Sasakian statistical manifolds.

An intrinsic characterization of isometric pluriharmonic immersions with codimension one

We give an intrinsic characterization of isometric pluriharmonic immersions of Kähler manifolds into semi-Euclidean spaces with real codimension one, which is a generalization of the Ricci-Curbastro theorem.