Hardy inequalities on Riemannian manifolds and applications

We prove a simple sufficient criteria to obtain some Hardy inequalities on Riemannian manifolds related to quasilinear second-order differential operator ${\mathrm{\Delta }}_{p}u:=\text{Div}(\text{abs}{\mathrm{\nabla }u}^{p-2}\mathrm{\nabla }u)$. Namely, if $\rho$ is a nonnegative weight such that $-{\mathrm{\Delta }}_p\rho \ge 0$, then the Hardy inequality $$c{\int }_M\frac{\text{abs}{u}^p}{{\rho }^p}\text{abs}{\mathrm{\nabla }\rho }^{p}d{v}_g\le {\int }_M\text{abs}{\mathrm{\nabla }u}^{p}d{v}_g,u\in {\text{Cinfinito}}_0(M)$$ holds. We show concrete examples specializing the function $\rho$.