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

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