Doubly Warped Product Cr-submanifolds in a Locally Conformal Kaehler Space Form

Recently, the present authors considered doubly warped product CR-submanifolds in a locally conformal Kaehler manifold and got some inequalities about the length of the second fundamental form ([14]). In this report, we obtain an inequality of the mean curvature of a doubly warped product CR-submanifold in a locally conformal Kaehler space form. Then, we consider the equality case of this inequality. 0. Introduction. On 1976, I. Vaisman redefined the notion of the locally conformal almost Kaehler structure on Hermitian manifolds ([18, 19]). Then, T. Kashiwada charecterised this notion by the tensor representation, and she gave the tensor representation of the curvature tensor of a locally conformal Kaehler space form under a certain condition ([9]). On the other hand, on 1978, A. Bejancu introduced the notion of CR-submanifolds which is a generalization of holomorphic and totally real submanifolds in an almost Hermitian manifold. After his definition, we can see many papers and books in this field ([6, 7, 12, 17, 20] etc.). Next, B. Y. Chen defined the notion of warped product CR-submanifolds in Kaehler manifolds and he proved a lot of interesting results in these submanifolds ([7]). This notion was considered in other Hermitian manifolds and gave similar results ([4]). Then we can also see the similar notion in an (almost) contact metric manifolds ([11, 20] etc.). Now, we can find only few papers about doubly warped product Riemannian manifolds which are the generalization of a warped product Riemannian manifold ([1, 8, 10]). 2000 Mathematics Subject Classification. 53C40.