Warped Product Submanifolds in Quaternion Space Forms

B.Y. Chen [3] 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. For a survey on warped product submanifolds we refer to [4]. In [8], we established a similar relationship between the warping function f (intrinsic structure) and the squared mean curvature and the holomorphic sectional curvature (extrinsic structures) for warped product submanifolds M1 ×f M2 in any complex space form. In the present paper, we investigate warped product submanifolds in quaternion space forms M̃(4c). We obtain several estimates of the mean curvature in terms of the warping function, whether c < 0, c = 0 and c > 0, respectively. Equality cases are considered and certain examples are given. As applications, we derive obstructions to minimal warped product submanifolds in quaternion space forms. As an example, the non-existence of minimal proper warped product submanifolds M1 ×f M2 in the m-dimensional quaternion Euclidean space Q with M1 compact is proved. 2000 Mathematics Subject Classification: 53C40, 53C25, 53C42. 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 important problems in the theory of submanifolds is the immersibility (or non-immersibility) of a Riemannian manifold in a Euclidean Supported by the CNCSIS grant 886 / 2005.