On Bach-flat gradient shrinking Ricci solitons

Abstract

In this article, we classify $n$-dimensional ($n\ge 4$) 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 ${\mathbb{R}}^4$ or the round cylinder ${\mathbb{S}}^3\times \mathbb{R}$. More generally, for $n\ge 5$, a Bach-flat gradient shrinking Ricci soliton is either Einstein, or a finite quotient of the Gaussian shrinking soliton ${\mathbb{R}}^n$ or the product $N^{n-1}\times \mathbb{R}$, where $N^{n-1}$ is Einstein.