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Article History

Received: 22 August 2023

Revised: 21 December 2023

Accepted: 14 October 2024

First Online: 10 November 2024

Declarations

:

: The authors declare no Conflict of interest.

: This article shows that a complete and connected Riemannian manifold can turn compact under specific conditions. We have also found an upper bound of the diameter for such a manifold. Later, triviality and integral requirements are established for a finite-volume non-compact complete quasi-Einstein manifold. It is shown that a complete Riemannian manifold has a finite fundamental group under certain restrictions. Some compactness criterion requirements have also been determined.