Rigidity of Gradient Shrinking Ricci Solitons

Abstract

We prove that a gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is rigid. For the $4$-dimensional case, we show that any gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is either Einstein, or a finite quotient of the Gaussian shrinking soliton ${\mathbb{R}}^4$, ${\mathbb{R}}^2\times {\mathbb{S}}^2$ or the round cylinder $\mathbb{R}\times {\mathbb{S}}^3$. Under the condition of fourth order divergence-free Weyl tensor, we have the same results.

26 pages

Cited by in corpus (1)

• On the classification of noncompact steady quasi-Einstein manifold with vanishing condition on the Weyl tensor