On Submersion of CR-Submanifolds of l.c.q.K. Manifold

We study submersion of CR-submanifolds of an l.c.q.K. manifold. We have shown that if an almost Hermitian manifold admits a Riemannian submersion of a CR-submanifold of a locally conformal quaternion Kaehler manifold , then is a locally conformal quaternion Kaehler manifold. 1. Introduction The concept of locally conformal Kaehler manifolds was introduced by Vaisman in [1]. Since then many papers appeared on these manifolds and their submanifolds (see [2] for details). However, the geometry of locally conformal quaternion Kaehler manifolds has been studied in [2–4] and their QR-submanifolds have been studied in [5]. A locally conformal quaternion Kaehler manifold (shortly, l.c.q.K. manifold) is a quaternion Hermitian manifold whose metric is conformal to a quaternion Kaehler metric in some neighborhood of each point. The main difference between locally conformal Kaehler manifolds and l.c.q.K. manifolds is that the Lee form of a compact l.c.q.K. manifold can be chosen as parallel form without any restrictions [2]. The study of the Riemannian submersion of a Riemannian manifold onto a Riemannian manifold was initiated by O’Neill [6]. A submersion naturally gives rise to two distributions on called the horizontal and vertical distributions, respectively, of which the vertical distribution is always integrable giving rise to the fibers of the submersion which are closed submanifold of . The notion of Cauchy-Riemann (CR) submanifold was introduced by Bejancu [7] as a natural generalization of complex submanifolds and totally real submanifolds. A CR-submanifolds of a l.c.q.K. manifold requires a differentiable holomorphic distribution , that is, for all , whose orthogonal complement is totally real distribution on , that is, for all . A CR-submanifold is called holomorphic submanifold if dim , totally real if dim and proper if it is neither holomorphic nor totally real. A CR-submanifold of a l.c.q.K. manifold is called a CR-product if it is Riemannian product of a holomorphic submanifold and a totally real submanifold of . Kobayashi [8] has proved that if an almost Hermitian manifold admits a Riemannian submersion of a CR-submanifold of a Kaehler Manifold , then is a Kaehler manifold. However, Deshmukh et al. [9] studied similar type of results for CR-submanifolds of manifolds in different classes of almost Hermitian manifolds, namely, Hermitian manifolds, quasi-Kaehler manifolds, and nearly Kaehler manifolds. In the present paper, we investigate submersion of CR-submanifold of a l.c.q.K. manifold and prove that if an almost Hermitian manifold admits a