A note on generalized Myers-type theorems for h-almost Ricci tensors and generalized quasi-Einstein tensors

TL;DR

This paper extends classical compactness theorems like Myers, Ambrose, and Galloway to Riemannian manifolds involving h-almost Ricci tensors and generalized quasi-Einstein tensors, including cases where h grows linearly.

Contribution

It introduces generalized Myers-type theorems for h-almost Ricci and quasi-Einstein tensors, broadening the scope of classical results in Riemannian geometry.

Findings

Proved compactness theorems for manifolds with h-almost Ricci tensors.

Extended theorems to cases where h has linear growth.

Generalized classical results to broader tensor conditions.

Abstract

In this paper, we prove some compactness theorems of Myers, Ambrose, and Galloway for complete Riemannian manifold in the concept of $h$-almost Ricci tensors and generalized quasi-Einstein tensors. Also, we extend the previous theorems when $h$ has at most linear growth in the distance function.