Contact Cr-Warped Product Submanifolds in Kenmotsu Space Forms

Another class of warped product CR-submanifolds in Kenmotsu manifolds

“…They establish an inequality for such type of warped products. In this paper, we study another type of warped products by reversing two factors which have not been considered in [1]. First, we prove the following results for later use.…”

“…In [1], Arslan et al studied warped product contact CR-submanifolds of the form M T × f M ⊥ , called contact CR-warped products of a Kenmotsu manifold M, where M T and M ⊥ are invariant and antiinvariant submanifolds of M, respectively. They establish an inequality for such type of warped products.…”

“…Since e * q+1 = ξ and for a warped product Riemannian submanifold M of a Kenmotsu manifold M, (ξ ln f) = 1 [1,17]. Then the above inequality will be…”

Semi-invariant warped product submanifolds of almost contact manifolds

“…Finally, if equality holds identically then from Equation ( 4 Semi-invariant warped product submanifolds of a generalized Sasakian space-form Hesigawa and Mihai [9] obtained the inequality for squared norm of second fundamental form for contact CR-warped product submanifolds in the setting of Sasakian space form. In the available literature, similar estimates are proved for squared norm of second fundamental form in contact manifolds (c.f., [3,4]). Since generalized Sasakian space form include the class of all almost contact metric manifold, so in this section we will obtain an estimate for the squared norm of second fundamental form for semi-invariant warped product submani-folds in the setting of generalized Sasakian space form.…”

“…With regard to physical applications of these manifolds, one may realize that space time around a massive star or a black hole can be modeled on a warped product manifolds for instance and warped product manifolds are widely used in differential geometry, Physics and as well as in different branches of Engineering. Due to wide applications of warped product submanifolds, this becomes a fascinating and interesting topic for research, and many articles are available in literature (c.f., [2][3][4]). CR-warped product was introduced by Chen [5].…”