Ridigity of Ricci Solitons with Weakly Harmonic Weyl Tensors

TL;DR

This paper establishes rigidity results for gradient shrinking Ricci solitons with weakly harmonic Weyl tensors, showing they are either Einstein or quotients of Einstein manifolds times Euclidean space.

Contribution

It generalizes previous results by proving rigidity under weaker conditions on the Weyl tensor for both compact and noncompact Ricci solitons.

Findings

Compact solitons are Einstein under the weak harmonic Weyl condition.

Noncompact solitons are rigid, being quotients of Einstein manifolds times Euclidean space.

Extends known rigidity results to broader classes of Ricci solitons.

Abstract

In this paper, we prove rigidity results on gradient shrinking Ricci solitons with weakly harmonic Weyl curvature tensors. Let $(M^n, g)$ be a compact gradient shrinking Ricci soliton satisfying ${\rm Ric}_g + Ddf = \rho g$ with $\rho >0$ constant. We show that if $(M,g)$ satisfies $\delta \mathcal W (\cdot, \cdot, \nabla f) = 0$, then $(M, g)$ is Einstein. Here $\mathcal W$ denotes the Weyl curvature tensor. In the case of noncompact, if $M$ is complete and satisfies the same condition, then $M$ is rigid in the sense that $M$ is given by a quotient of product of an Einstein manifold with Euclidean space. These are generalizations of the previous known results in \cite{l-r}, \cite{m-s} and \cite{p-w3}.