Bi-warped Product Submanifolds of 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 form M=MT×f1M⊥×f2Mθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M=M_T\times _{f_1} M_\perp \times _{f_2} M_\theta $$\end{document} in a nearly Kaehler manifold satisfies the following sharp inequality: ‖h‖2≥2p‖∇(lnf1)‖2+4q1+109cot2θ‖∇(lnf2)‖2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert h\Vert ^2\ge 2p\Vert \nabla (\ln f_1)\Vert ^2+4q\left( 1+{\frac{10}{9}}\cot ^2\theta \right) \Vert \nabla (\ln f_2)\Vert ^2, \end{aligned}$$\end{document}where p=dimM⊥\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=\dim M_\perp $$\end{document}, q=12dimMθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=\frac{1}{2}\dim M_\theta $$\end{document}, and f1,f2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_1,\,f_2$$\end{document} are smooth positive functions on MT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_T$$\end{document}. We also investigate the equality case of this inequality. Further, some applications of this inequality are also given.