» Warped product pointwise bi-slant submanifolds of trans-Sasakian manifold

The purpose of this paper is to study pointwise bi-slant submanifolds of trans-Sasakian manifold. Firstly, we obtain a non-trivial example of a pointwise bi-slant submanifolds of an almost contact metric manifold. Next we provide some fundamental results, including a characterization for warped product pointwise bi-slant submanifolds in trans-Sasakian manifold. Then we establish that there does not exist warped product pointwise bi-slant submanifold of trans-Sasakian manifold $\tilde{M}$ under some certain considerations. Next, we consider that $M$ is a proper pointwise bi-slant submanifold of a trans-Sasakian manifold $\tilde{M}$ with pointwise slant distrbutions $\mathcal{D}_1\oplus<\xi>$ and $\mathcal{D}_2$ , then using Hiepko’s Theorem, $M$ becomes a locally warped product submanifold of the form $M_1\times_fM_2$ , where $M_1$ and $M_2$ are pointwise slant submanifolds with the slant angles $\theta_1$ and $\theta_2$ respectively. Later, we show that pointwise bi-slant submanifolds of trans-Sasakian manifold become Einstein manifolds admitting Ricci soliton and gradient Ricci soliton under some certain conditions..