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**Warped product CR-submanifolds in locally conformal Kähler manifolds.**
*(English)*
Zbl 1104.53049

Summary: Recently, B. Y. Chen introduced the notion of warped product CR-submanifolds and CR-warped products of Kähler manifolds, that is, a warped product Riemannian submanifold of a holomorphic submanifold and a totally real submanifold in a Kähler manifold, cf. B. Y. Chen [Monatsh. Math. 133, 177–195 (2001; Zbl 0996.53044)]. In this paper we find a lot of essential and interesting properties of these submanifolds.

We research such submanifolds in locally conformal Kähler manifolds.

There are two types of warped product CR-submanifolds. One of them is not interesting to us, as it becomes trivial under a certain condition. We shall concentrate on another type (we call it a CR-warped product). In a CR-warped product in an l.c. K.-manifold, we prove an inequality. Next, we consider the equality case and we show that some anti-holomorphic CR-warped product satisfying a certain condition in an l.c. K.-manifold satisfy the equality (see Theorem 4.3). Finally, in a proper CR-warped product which satisfies the equality, we prove that its holomorphic submanifold in an l.c. K.-space form.

We research such submanifolds in locally conformal Kähler manifolds.

There are two types of warped product CR-submanifolds. One of them is not interesting to us, as it becomes trivial under a certain condition. We shall concentrate on another type (we call it a CR-warped product). In a CR-warped product in an l.c. K.-manifold, we prove an inequality. Next, we consider the equality case and we show that some anti-holomorphic CR-warped product satisfying a certain condition in an l.c. K.-manifold satisfy the equality (see Theorem 4.3). Finally, in a proper CR-warped product which satisfies the equality, we prove that its holomorphic submanifold in an l.c. K.-space form.