The Ricci tensor of a gradient Ricci soliton with harmonic Weyl tensor
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918160070868992 |
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| author | Borges, Valter Horácio, Matheus Andrade Ribeiro de Moura Santos, João Paulo dos |
| author_facet | Borges, Valter Horácio, Matheus Andrade Ribeiro de Moura Santos, João Paulo dos |
| contents | 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 $n \geq 4$ 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 $n\geq5$. Our proof offers two notable advantages: it is shorter and does not require the use of any specialized moving frame. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11939 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Ricci tensor of a gradient Ricci soliton with harmonic Weyl tensor Borges, Valter Horácio, Matheus Andrade Ribeiro de Moura Santos, João Paulo dos Differential Geometry 53C20, 53E20, 53C21, 53C25 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 $n \geq 4$ 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 $n\geq5$. Our proof offers two notable advantages: it is shorter and does not require the use of any specialized moving frame. |
| title | The Ricci tensor of a gradient Ricci soliton with harmonic Weyl tensor |
| topic | Differential Geometry 53C20, 53E20, 53C21, 53C25 |
| url | https://arxiv.org/abs/2510.11939 |