Weyl scalars on compact Ricci solitons

TL;DR

This paper studies conditions involving the Weyl tensor that ensure compact Ricci solitons are Einstein, advancing the classification of Ricci solitons through new scalar curvature conditions.

Contribution

It introduces a general scalar condition involving the Weyl tensor that guarantees a compact Ricci soliton is Einstein, improving and unifying previous results.

Findings

A generic linear combination of divergences of the Weyl tensor implies Einstein condition.

The results recover and improve known classifications of Ricci solitons.

Proposes a program to classify Ricci solitons via vanishing conditions on Weyl tensor derivatives.

Abstract

We investigate the triviality of compact Ricci solitons under general scalar conditions involving the Weyl tensor. More precisely, we show that a compact Ricci soliton is Einstein if a generic linear combination of divergences of the Weyl tensor contracted with suitable covariant derivatives of the potential function vanishes. In particular we recover and improve all known related results. This paper can be thought as a first, preliminary step in a general program which aims at showing that Ricci solitons can be classified finding a "generic" $[k, s]$-vanishing condition on the Weyl tensor, for every $k, s\in\mathbb{N}$, where $k$ is the order of the covariant derivatives of Weyl and $s$ is the type of the (covariant) tensor involved.