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

TL;DR

This paper investigates the geometric properties of $(m, ho)$-quasi Einstein manifolds, establishing conditions for compactness, diameter bounds, and fundamental group finiteness, thereby advancing understanding of their structure.

Contribution

It provides new diameter bounds, compactness criteria, and relations to the Hodge-de Rham potential for $(m, ho)$-quasi Einstein manifolds, extending previous geometric analysis.

Findings

Complete and connected manifolds become compact under certain conditions.

An upper bound for the diameter of such manifolds is determined.

Conditions for finite fundamental group and compactness are established.

Abstract

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