Gradient Ricci solitons admitting a closed conformal Vector field

Abstract

We prove that a gradient Ricci soliton (Mn,g) admitting a nonparallel closed conformal vector field must be locally conformally flat, provided that the dimension is 3 or 4. Moreover, in the dimension n≥5, we show that a gradient Ricci soliton admitting a nonparallel closed conformal vector field must have harmonic Weyl tensor. In particular, we conclude that a gradient shrinking Ricci soliton admitting a nonparallel closed conformal vector field must be rigid.