Rigidity Results on Generalized m-Quasi Einstein Manifolds with Associated Affine Killing Vector Field

We study a non-trivial generalized m-quasi Einstein manifold M with finite m and associated divergence-free affine Killing vector field, and show that M reduces to an m-quasi Einstein manifold. In addition, if M is complete, then it splits as the product of a line and an (n - 1)dimensional negatively Einstein manifold. Finally, we show that the same result holds for a complete non-trivial m-quasi Einstein manifold M with finite m and associated affine Killing vector field.