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On Bach-flat gradient shrinking Ricci solitons. (English) Zbl 1277.53036
A complete Riemannian manifold $$(M^n,g_{ij})$$ is called a gradient Ricci soliton if there exists a smooth function $$f$$ on $$M^n$$ such that the Ricci tensor $$R_{ij}$$ of the metric $$g$$ satisfies the equation $R_{ij}+\nabla_i\nabla_j f=\rho g_{ij}$ for some constant $$\rho$$. For $$\rho=0$$ the Ricci soliton is steady, for $$\rho>0$$ it is shrinking, and for $$\rho<0$$ expanding. In this paper, the authors investigate an interesting class of complete gradient shrinking Ricci solitons: those with vanishing Bach tensor. On any manifold $$(M^n,g_{ij})$$, $$n\geqslant 4$$, the Bach tensor is defined by $B_{ij}=\frac{1}{n-3}\nabla^{k}\nabla^{l} W_{ikjl}+\frac{1}{n-2}R_{kl}W_{i_j}{}^{k l},$ $$W_{ikjl}$$ being the Weyl tensor. It is easy to see that if $$(M^n,g_{ij})$$ is either locally conformally flat or Einstein, then $$(M^n,g_{ij})$$ is Bach-flat: $$B_{ij}=0$$. The authors show that Bach-flat 4-dimensional gradient shrinking Ricci solitons are either Einstein or locally conformally flat. More generally, for $$n\geqslant 5$$, they prove that 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 an Einstein manifold of positive scalar curvature.

##### MSC:
 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 53C20 Global Riemannian geometry, including pinching 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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