Geometry of bi-warped product submanifolds in Sasakian and cosymplectic manifolds

A bi-warped product of the form: M=NT×f1N⊥n1×f2Nθn2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M=N_T \times _{f_1}N^{n_{1}}_\perp \times _{f_2} N^{n_{2}}_\theta $$\end{document} in a contact metric manifold is called a CRS bi-warped product, where NT,N⊥n1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_T,\, N^{n_{1}}_\perp $$\end{document} and Nθn2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^{n_{2}}_\theta $$\end{document} are invariant, anti-invariant and proper pointwise slant submanifolds, respectively. First, we prove that there are no proper CRS bi-warped products other than contact CR-biwarped products in any Sasakian manifold. Then, we prove that if M is a CRS bi-warped product in a cosymplectic manifold, its second fundamental form h satisfies ‖h‖2≥2n1‖∇(lnf1)‖2+2n2(1+2cot2θ)‖∇(lnf2)‖2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert h\Vert ^2\ge 2n_1\Vert \nabla (\ln f_1)\Vert ^2+2n_2(1+2\cot ^2\theta )\Vert \nabla (\ln f_2)\Vert ^2. \end{aligned}$$\end{document}Several applications of this inequality are given. Finally, we provide a non-trivial example of CRS bi-warped product which satisfies the equality case.