Izumi Hasegawa

Contact CR-Warped Product Submanifolds in Sasakian Manifolds

Recently, B.-Y. Chen studied warped products which are CR-submanifolds in Kaehler manifolds and established general sharp inequalities for CR-warped products in Kaehler manifolds. In the present paper, we obtain a sharp inequality for the squared norm of the second fundamental form (an extrinsic invariant) in terms of the warping function for contact CR-warped products isometrically immersed in Sasakian manifolds. The equality case is considered. Also, the minimum codimension of a contact CR-warped product in an odd-dimensional sphere is determined.

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.

Sasakian Structures on Finsler Manifolds

Let M be an (m +1)-dimensional Finsler manifold with fundamental function L satisfying

INFINITESIMAL PROJECTIVE TRANSFORMATIONS ON TANGENT BUNDLES WITH LIFT CONNECTIONS

L(x,ky) = kL(x,y),k >0

On Sasakian manifolds with vanishing contact Bochner curvature tensor II

(1.1) and

Randers spaces of constant flag curvature induced by almost contact metric structures

rank\left( {{g_{ij}}} \right) = m + 1