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.
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Acknowledgements
The authors are supported by the GNAMPA project “Strutture di tipo Einstein e Analisi Geometrica su varietà Riemanniane e Lorentziane”. The first author is supported also by the PRIN Project “Variational methods, with applications to problems in mathematical physics and geometry”.
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Catino, G., Mastrolia, P. Weyl Scalars on Compact Ricci Solitons. J Geom Anal 29, 3328–3344 (2019). https://doi.org/10.1007/s12220-018-00120-z
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DOI: https://doi.org/10.1007/s12220-018-00120-z