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

Bonanzinga, Vittoria; Matsumoto, Koji

Period. Math. Hung. 48, No. 1-2, 207-221 (2004).

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.

MSC:

53C40	Global submanifolds
53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)

Keywords:

locally conformal Kähler manifold; locally conformal Kähler space form; CR-submanifold; warped product manifold; anti-holomorphic submanifold

Citations:

Zbl 0996.53044