2025/05/09 by Sharma, Ramesh
#53C21 #53C25 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2505.06140
We extend the following result of Cochran ``A closed m-quasi Einstein manifold (M,g,X) with m ≠ -2 has constant scalar curvature if and only if X is Killing" covering the missing accidental case m=-2 and generalize it showing that X is Killing if the integral of the Lie derivative of the scalar curvature along X is non-positive. For a closed m-quasi Einstein manifold of dimension n ≥ 2, if X is conformal, then it is Killing; and in addition, if M admits a non-Killing conformal vector field V, then it is globally isometric to a sphere and V is gradient for n > 2. Finally, we derive an integral identity for a vector field on a closed Riemannian manifold, which provides a direct proof of the Bourguignon-Ezin conservation identity.