ON TWO DIFFERENT CLASSES OF WARPED PRODUCT SUBMANIFOLDS OF KENMOTSU MANIFOLDS

Warped product skew CR-submanifold of the form M = M1 xf M1 of a Kenmotsu manifold M over bar (throughout the paper), where M1 = MT x M theta and MT , M1, M theta represents invariant, anti-invariant and proper slant submanifold of over bar M, studied in [28] and another class of warped product skew CR-submanifold of the form M = M2 xf MT of over bar M, where M2 = M1 x M theta is studied in [19]. Also the warped product submanifold of the form M = M3 xf M theta of over bar M, where M3 = MT x M1 and MT , M1, M theta represents invariant, anti-invariant and proper point wise slant submanifold of over bar M, were studied in [18]. As a generalization of the above mentioned three classes, we consider a class of warped product submanifold of the form M = M4 xf M theta 3 of over bar M, where M4 = M theta 1 x M theta 2 in which M theta 1 and M theta 2 are proper slant submanifolds of M over bar and M theta 3 represents a proper pointwise slant submanifold of over bar M. A characterization is given on the existence of such warped product submanifolds which generalizes the characterization of warped product submanifolds of the form M = M1 xf M1, studied in [28], the characterization of warped product submanifolds of the form M = M2 xf MT, studied in [19], the characterization of warped product submanifolds of the form M = M3 xf M theta, studied in [18] and also the characterization of warped product pointwise bi-slant submanifolds of over bar M, studied in [17]. Since warped product bi-slant submanifolds of M over bar does not exist (Theorem 4.2 of [17]), the Riemannian product M4 = M theta 1 x M theta 2 cannot be a warped product. So, for studying the bi-warped product submanifolds of M over bar of the form M theta 1 xf1 M theta 2 xf2 M theta 3, we have taken M theta 1, M theta 2, M theta 3 as pointwise slant submanifolds of M over bar of distinct slant functions theta 1, theta 2, theta 3 respectively. The existence of such type of bi-warped product submanifolds of M over bar is ensured by an example. Finally, a Chen-type inequality on the squared norm of the second fundamental form of such bi-warped product submanifolds of M over bar is obtained which also generalizes the inequalities obtained in [33], [18] and [17], respectively.