Some generalized inequalities in Riemannian Geometry

[1] investigates advanced connotations of Hardy and Rellich-type inequalities on complete noncompact Riemannian manifolds, delving on deriving inequalities that incorporate poignant weight functions. These inequalities prolongate classical results by providing sharper estimates conforming the geometry and structure of the underlying manifold. We in this paper further extend the inequalities and in due course exhibit them in section [4] in terms of theorems and proofs.