We prove some isoperimetric type inequalities in warped product manifolds, or more generally, multiply manifolds. then relate them to involving the higher order mean-curvature integrals. also apply our results obtain sharp eigenvalue estimates and geometric space forms.

In this paper, we consider Serrin’s overdetermined problems in warped product manifolds and prove type rigidity results by using the [Formula: see text]-function approach introduced Weinberger.

We find the necessary and sufficient conditions for a sequential warped product manifold to be quasi-Einstein manifold. also investigate standard static space-time generalized Robertson-Walker of quasi-constant curvature.