On Bach-flat gradient shrinking Ricci solitons

In this article, we classify n-dimensional (n≥4) complete Bach-flat gradient shrinking Ricci solitons. More precisely, we prove that any 4-dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat and hence a finite quotient of the Gaussian shrinking soliton R4 or the round cylinder S3×R. More generally, for n≥5, a Bach-flat gradient shrinking Ricci soliton is either Einstein, or a finite quotient of the Gaussian shrinking soliton Rn or the product Nn−1×R, where Nn−1 is Einstein.