Almost Ricci solitons on Finsler spaces
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909380118577152 |
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| author | Xia, Qiaoling |
| author_facet | Xia, Qiaoling |
| contents | In this paper, (gradient) almost Ricci solitons on Finsler measure spaces $(M, F, m)$ are introduced and investigated. We prove that $(M, F, m)$ is a gradient almost Ricci soliton if and only if the infinity-Ricci curvature Ric$_\infty$ is a scalar function on $M$ when $M$ is compact. Moreover, we give an equivalent characterization of (gradient) almost Ricci solitons for Randers metrics $F=α+β$, which implies that every Randers (gradient) almost Ricci soliton is of isotropic S$_{BH}$-curvature. Based on this and the navigation technique, we further classify Randers almost Ricci solitons (resp. gradient almost Ricci solitons) up to classifications of Randers Einstein metrics $F$ (resp. Riemannian gradient almost Ricci solitons) and the homothetic vector fields of $F$ (resp. solutions of the equation which the weight function $f$ of $m$ satisfies) when $F$ has isotropic S$_{BH}$-curvature. As applications, we obtain some rigidity results for compact Randers (gradient) Ricci solitons and construct several Randers gradient Ricci solitons, which are the first nontrivial examples of gradient Ricci solitons in Finsler geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02038 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Almost Ricci solitons on Finsler spaces Xia, Qiaoling Differential Geometry In this paper, (gradient) almost Ricci solitons on Finsler measure spaces $(M, F, m)$ are introduced and investigated. We prove that $(M, F, m)$ is a gradient almost Ricci soliton if and only if the infinity-Ricci curvature Ric$_\infty$ is a scalar function on $M$ when $M$ is compact. Moreover, we give an equivalent characterization of (gradient) almost Ricci solitons for Randers metrics $F=α+β$, which implies that every Randers (gradient) almost Ricci soliton is of isotropic S$_{BH}$-curvature. Based on this and the navigation technique, we further classify Randers almost Ricci solitons (resp. gradient almost Ricci solitons) up to classifications of Randers Einstein metrics $F$ (resp. Riemannian gradient almost Ricci solitons) and the homothetic vector fields of $F$ (resp. solutions of the equation which the weight function $f$ of $m$ satisfies) when $F$ has isotropic S$_{BH}$-curvature. As applications, we obtain some rigidity results for compact Randers (gradient) Ricci solitons and construct several Randers gradient Ricci solitons, which are the first nontrivial examples of gradient Ricci solitons in Finsler geometry. |
| title | Almost Ricci solitons on Finsler spaces |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2403.02038 |