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This paper aims to study the $(m,\rho)$-quasi Einstein manifold. This articleshows that a complete and connected Riemannian manifold under certainconditions becomes compact. Also, we have determined an upper bound of thediameter for such a manifold. It is also exhibited that the potential functionacquiesces to the Hodge-de Rham potential up to a real constant in an$(m,\rho)$-quasi Einstein manifold. Later, some triviality and integralconditions are established for a non-compact complete $(m,\rho)$-quasi Einsteinmanifold having finite volume. Finally, it is proved that with some certainconstraints, a complete Riemannian manifold admits finite fundamental group.Furthermore, some conditions for compactness criteria have also been deduced.

Diameter estimation of $(m,\rho)$-quasi Einstein manifolds