Gradient Ricci solitons with vanishing conditions on Weyl

TL;DR

This paper classifies complete gradient Ricci solitons with specific vanishing conditions on the Weyl tensor, revealing their structure as Einstein or product manifolds, and extends results to steady and expanding cases across all dimensions.

Contribution

It provides a classification of gradient Ricci solitons with higher-order divergence-free Weyl tensor conditions, including new results in three dimensions and for steady and expanding cases.

Findings

Gradient shrinking Ricci solitons are either Einstein or a product with Gaussian solitons.

Three-dimensional steady solitons with vanishing double divergence of the Bach tensor are flat or Bryant solitons.

The technique applies to steady and expanding cases in all dimensions.

Abstract

We classify complete gradient Ricci solitons satisfying a fourth-order vanishing condition on the Weyl tensor, improving previously known results. More precisely, we show that any $n$-dimensional ($n\geq 4$) gradient shrinking Ricci soliton with fourth order divergence-free Weyl tensor is either Einstein, or a finite quotient of $N^{n-k}\times \mathbb{R}^k$, $(k > 0)$, the product of a Einstein manifold $N^{n-k}$ with the Gaussian shrinking soliton $\mathbb{R}^k$. The technique applies also to the steady and expanding cases in all dimensions. In particular, we prove that a three dimensional gradient steady soliton with third order divergence-free Cotton tensor, i.e. with vanishing double divergence of the Bach tensor, is either flat or isometric to the Bryant soliton.