Geometry of bi-warped product submanifolds in Sasakian manifolds.

In this paper, we study bi-warped product submanifolds of Sasakian manifolds, which are the natural generalizations of single warped products and Remannian products. We show that if $M$ is a bi-warped product submanifold of the form $M=N_T \times_{f_1}N_\perp\times_{f_2} N_\theta$ of a Sasakian manifold $\tilde M$, where $N_T,\, N_\perp$ and $N_\theta$ are invariant, anti-invariant and proper pointwise slant submanifolds of $\tilde M$, respectively then the second fundamental form of $M$ satisfies a general inequality: $\|h\|^2\geq 2n_1(\|\vec\nabla(\ln f_1)\|^2+1)+2n_2(1+2\cot^2\theta)\|\vec\nabla(\ln f_2)\|^2+2n_2(1+\cos^4\theta)$, where $n_1=\dim(N_\perp),\,n_2=\dim(N_\theta)$ and $h$ is the second fundamental form and $\vec\nabla(\ln f_1)$ and $\vec\nabla(\ln f_2)$ are the gradient components along $N_\perp$ and $N_\theta$, respectively. Some applications of this inequality are given and we provide some non-trivial examples of bi-warped products.