The Ricci tensor of a gradient Ricci soliton with harmonic Weyl tensor

2025/10/13 by Borges, Valter, Horácio, Matheus Andrade Ribeiro de Moura, Santos, João Paulo dos · 2 citations
#53C20 #53C21 #53C25 #53E20 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.11939

Abstract

In this article, we give a new proof of a result due to J. Kim, which states that the Ricci tensor of a gradient Ricci soliton with dimension n ≥ 4 and harmonic Weyl tensor has at most three distinct eigenvalues. This result constitutes an essential step in the classification of such manifolds, originally established by J. Kim in dimension 4 and subsequently extended to dimensions n≥5. Our proof offers two notable advantages: it is shorter and does not require the use of any specialized moving frame.

Citations

• Classification of Gradient Ricci solitons with harmonic Weyl curvature
• Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature
• On a classification of 4-d gradient Ricci solitons with harmonic Weyl curvature
• On Bach-flat gradient shrinking Ricci solitons
• On gradient Ricci solitons
• Some New Examples of Non-Kähler Ricci Solitons

Cited by

• Quasi-Einstein manifolds with Harmonic Weyl curvature
• Local Structure of Gradient Almost Ricci Solitons with Harmonic Weyl Tensor

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