A note on closed quasi-Einstein manifolds

The notion of m-quasi-Einstein manifolds originates from the study of Einstein warped product metrics and they are influential in constructing for many physical models. For example, these manifolds arises for extremal isolated horizons in the theory of black holes. In a recent work by Cochran (arXiv:2404.17090v1, 2024), the author studied Killing vector fields on closed m-quasi-Einstein manifolds. In this short paper, we will give another proof of his main result involving the scalar curvature, which holds for all values of m and is based on the use of known formulae related to quasi-Einstein metrics.