Geometric inequalities of $\mathcal{P}\mathcal{R}$-warped product submanifold in para-Kenmotsu manifold

The main purpose of this paper is to study the properties of $\mathcal{P}\mathcal{R}$-semi-invariant submanifold of para-Kenmotsu manifold. We obtain the integrability conditions for the invariant distribution and anti-invariant distribution. We obtain some existence and non-existence results of $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifolds. We provide some necessary and sufficient conditions for $\mathcal{P}\mathcal{R}$-semi-invariant submanifold to be a $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifold in para-Kenmotsu manifold. We also derive some sharp inequalities for $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifold in para-Kenmotsu manifolds.