Rigid properties for gradient generalized m-quasi-Einstein manifolds and gradient shrinking Ricci solitons

Abstract

In this paper, we discuss the rigidity of gradient generalized m-quasi-Einstein manifolds and gradient shrinking Ricci solitons. We get some useful equations for the potential function of gradient generalized m-quasi-Einstein manifolds. Using these equations, we obtain some rigid properties for gradient generalized m-quasi-Einstein manifolds. Furthermore, we prove that some complete gradient shrinking Ricci solitons with pinching condition on the Weyl tensor are finite quotients of Rn or Sn.