Kenmotsu statistical manifolds and warped product

Abstract

A notion of a Kenmotsu statistical manifold is introduced, which is locally obtained as the warped product of a holomorphic statistical manifold and a line. A statistical manifold is a Riemannian manifold equipped with a torsion-free affine connection satisfying the Codazzi equation. It can be considered as being in information geometry, Hessian geometry and various submanifold theory. On the other hand, a Kenmotsu manifold is in a meaningful class of almost contact metric manifolds. In this paper, we construct a suitable statistical structure on it. Although the notion of the warped product of Riemannian manifolds is well known, the one for statistical manifolds is not established. We consider it in general, and study the statistical sectional curvature of the warped product of two statistical manifolds. We show that a Kenmotsu statistical manifold of constant $\phi$-sectional curvature is constructed from a special Kähler manifold, which is an important example of holomorphic statistical manifold. A Sasakian statistical manifold is also studied from the viewpoint of the warped product of statistical manifolds.