Rigidity of gradient shrinking Ricci solitons

We prove that an $n$-dimensional $(n \geq 4)$ gradient shrinking Ricci soliton with fourth-order divergence free Riemannian curvature tensor (i.e. $\mathit{\operatorname{div}}^4 Rm = 0)$ is rigid. In particular, such a soliton in dimension $4$ is either Einstein, or a finite quotient of $\mathbb{R}^4$, $\mathbb{R}^2 \times \mathbb{S}^2$, or $\mathbb{R} \times \mathbb{S}^3$. Under the condition of $\mathit{\operatorname{div}}^3 W (\nabla f) = 0$, we have the same results.

Keywords

rigidity, Gradient shrinking Ricci soliton, Riemannian curvature tensor, Weyl curvature tensor

2010 Mathematics Subject Classification

53C24, 53C25

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This work is partially supported by Natural Science Foundation of China (No. 11601495) and Science Foundation for The Excellent Young Scholars of Central Universities (No. CUGL170213).

Received 18 October 2018

Accepted 4 October 2019

Published 18 February 2021