Warped Product Submanifolds of Kenmotsu Manifolds with Slant Fiber

Abstract

Recently, we have discussed the warped product pseudo-slant submanifolds of the type $M_\theta\times_fM_\perp$ of Kenmotsu manifolds. In this paper, we study other type of warped product pseudo-slant submanifolds by reversing these two factors in Kenmotsu manifolds. The existence of such warped product immersions is proved by a characterization. Also, we provide an example of warped product pseudo-slant submanifolds. Finally, we establish a sharp estimation such as $\|h\|^2\geq 2p\cos^2\theta\big(\|\vec\nabla\ln f\|^2-1\big)$ for the squared norm of the second fundamental form $\|h\|^2$, in terms of the warping function $f$, where $\vec\nabla\ln f$ is the gradient vector of the function $\ln f$. The equality case is also discussed.