Warped product semi-slant submanifolds in locally conformal Kaehler manifolds II

Abstract

In 1994 N.~Papaghiuc introduced the notion of semi-slant submanifold in a Hermitian manifold which is a generalization of $CR$- and slant-submanifolds, \cite{MR0353212}, \cite{MR760392}.
 In particular, he considered this submanifold in Kaehlerian manifolds, \cite{MR1328947}.
 Then, in 2007, V.~A.~Khan and M.~A.~Khan considered this submanifold in a nearly Kaehler manifold and obtained interesting results, \cite{MR2364904}.
 Recently, we considered semi-slant submanifolds in a locally conformal Kaehler manifold and we gave a necessary and sufficient conditions of the two distributions (holomorphic and slant) be integrable.
 Moreover, we considered these submanifolds in a locally conformal Kaehler space form.
 In the last paper, we defined $2$-kind warped product semi-slant submanifolds in almost hermitian manifolds and studied the first kind submanifold in a locally conformal Kaehler manifold.
 Using Gauss equation, we derived some properties of this submanifold in an locally conformal Kaehler space form, \cite{MR2077697}, \cite{MR3728534}.
 In this paper, we consider same submanifold with the parallel second fundamental form in a locally conformal Kaehler space form.
 Using Codazzi equation, we partially determine the tensor field $P$ which defined in~\eqref{1.3}, see Theorem~\ref{th4.1}.
 Finally, we show that, in the first type warped product semi-slant submanifold in a locally conformal space form, if it is normally flat, then the shape operators $A$
 satisfy some special equations, see Theorem~\ref{th5.2}.