$CR$-Submanifolds With the Symmetric $\nabla \sigma$ in a Locally Conformal Kaehler Space Form

Koji Matsumotoa

In this paper, we consider $CR$-submanifolds with the symmetric $\nabla\sigma$ which is a generalization of parallel second fundamental form, in a locally conformal Kaehler space form. About the symmetric tensor field $P$ defined in (1.7), we show that, in an anti-holomorphic submanifold in an l.c.K.-space form, $P$ is diagonal with respect to an adapted frame and has two eigenfunctions (See Theorem 3.1). Finally, we consider the relation of the eigenfunctions of $P$ and the Lee form (See Theorems 3.2 and 3.3).