GRADIENT ALMOST RICCI SOLITONS WITH VANISHING CONDITIONS ON WEYL TENSOR AND BACH TENSOR

Abstract

. In this paper we consider gradient almost Ricci solitons with weak conditions on Weyl and Bach tensors. We show that a gradient almost Ricci soliton has harmonic Weyl curvature if it has fourth order divergence-free Weyl tensor, or it has divergence-free Bach tensor. Fur- thermore, if its Weyl tensor is radially flat, we prove such a gradient almost Ricci soliton is locally a warped product with Einstein fibers. Fi- nally, we prove a rigidity result on compact gradient almost Ricci solitons satisfying an integral condition.