Warped Product Submanifolds in Generalized Complex Space Forms

B.Y. Chen [5] established a sharp inequality for the warping function of a warped product submanifold in a Riemannian space form in terms of the squared mean curvature. Later, in [4], he studied warped product submanifolds in complex hyperbolic spaces. In the present paper, we establish an inequality between the warping function f (intrinsic structure) and the squared mean curvature ‖H‖2 and the holomorphic sectional curvature c (extrinsic structures) for warped product submanifolds M1 ×f M2 in any generalized complex space form f M(c, α). Introduction The notion of warped product plays some important role in differential geometry as well as in physics [3]. For instance, the best relativistic model of the Schwarzschild space-time that describes the out space around a massive star or a black hole is given as a warped product. One of the most fundamental problems in the theory of submanifolds is the immersibility (or non-immersibility) of a Riemannian manifold in a Euclidean space (or, more generally, in a space form). According to a well-known theorem on Nash, every Riemannian manifold can be isometrically immersed in some Euclidean spaces with sufficiently high codimension. Nash’s theorem implies, in particular, that every warped product M1×f M2 can be immersed as a Riemannian submanifold in some Euclidean space. Moreover, many important submanifolds in real and complex space forms are expressed as a warped product submanifold. Every Riemannian manifold of constant curvature c can be locally expressed as a warped product whose warping function satisfies ∆f = cf . For example, S(1) is locally isometric to (−2 , π2 ) ×cos t Sn−1(1), E is locally isometric to (0,∞) ×x Sn−1(1) and Hn(−1) is locally isometric to R×ex En−1 (see [3]). 2000 Mathematics Subject Classification. 53C40, 53C15, 53C42.