Recently, Arslan et al. [K. Arslan, R. Ezentas, I. Mihai, C. Murathan, J. Korean Math. Soc., 42 (2005), 1101-1110] studied contact CR-warped product submanifolds of the form M T × f M ⊥ of a Kenmotsu manifold M, where M T and M ⊥ are invariant and anti-invariant submanifolds of M, respectively. In this paper, we study the warped product submanifolds by reversing these two factors, i.e., the warped products of the form M ⊥ × f M T which have not been considered in earlier studies. On the existence of such warped products, a characterization is given. A sharp estimation for the squared norm of the second fundamental form is obtained, and in the statement of inequality, the equality case is considered. Finally, we provide two examples of non-trivial warped product submanifolds.

We show in this paper that many well-known theorems about the geometry of warped product submanifolds of Kaehler manifolds and itself nearly Kaehler manifolds can be generalized to CR-slant warped products in nearly Kaehler manifolds.

We study bi-warped product submanifolds of nearly Kaehler manifolds which are the natural extension of warped products. We prove that every bi-warped product submanifold of the formKaehler manifold satisfies the following sharp inequality:where p = dim M ⊥ , q = 1 2 dim M θ , and f 1 , f 2 are smooth positive functions on M T . We also investigate the equality case of this inequality. Further, some applications of this inequality are also given.

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