Periodica Mathematica Hungarica Vol. 48 (1–2), 2004, pp. 207–221 WARPED PRODUCT CR-SUBMANIFOLDS IN LOCALLY CONFORMAL KAEHLER MANIFOLDS Vittoria Bonanzinga (Reggio Calabria) and Koji Matsumoto (Yamagata) Abstract Recently, B. Y. Chen introduced the notion of warped product CR-subman- ifolds and CR-warped products of Kaehler manifolds, that is, a warped product Riemannian submanifold of a holomorphic submanifold and a totally real subman- ifold in a Kaehler manifold ([C-3]). [C-3] follows [MM], [Mi]. In this paper we find a lot of essential and interesting properties of these submanifolds. In this paper, we research such submanifolds in locally conformal Kaehler manifolds. There are two types of warped product CR-submanifolds. But, one of them is not interest to us, as it becomes trivial under a certain condition. (See Theorem 2.2). So, 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 one inequality (See Theorem 4.1). 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 equal- ity (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 is also an l.c.K.-space form and its totally real submanifold is a real space form (See The- orems 4.6 and 4.7). 1. Locally conformal Kaehler manifolds and their submanifolds A Hermitian manifold ˜ M with structure (J, 〈 , 〉) is called a locally conformal Kaehler (an l.c.K.) manifold if each point x ∈ ˜ M has an open neighbourhood U with a differentiable function ρ : U → R such that 〈 , 〉 ∗ = e -2ρ 〈 , 〉| U is a Kaehlerian metric, that is, ∇ ∗ J = 0, where J is the almost complex structure and ∇ ∗ J is the covariant differentiation with respect to 〈 , 〉 ∗ . Then we have Mathematics subject classification number: 53C40, 53C25. Key words and phrases: l.c.K.-manifold, l.c.K.-space form, CR-submanifold, warped product manifold, anti-holomorphic submanifold. This paper was written while the second author visited University of Reggio Calabria, DIMET (Italy) as a visiting professor, supported by a GNSAGA research fellowship and Depart- ment of Geometry and Topology, Faculty of Science, University of Granada (Spain) as a visiting professor. He would like to express his hearty thanks for the hospitality received during his stay. 0031-5303/2004/$20.00 Akad´ emiai Kiad´o, Budapest c  Akad´ emiai Kiad´o, Budapest Kluwer Academic Publishers, Dordrecht