Almost Ricci Solitons on Class $\mathcal A$ Hypersurfaces of Product Spaces
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| Format: | Preprint |
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2025
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| author | Çoraplı, Ahmet Umut Demirci, Burcu Bektaş Turgay, Nurettin Cenk |
| author_facet | Çoraplı, Ahmet Umut Demirci, Burcu Bektaş Turgay, Nurettin Cenk |
| contents | In this paper, we study hypersurfaces in the product spaces $\mathbb{Q}_ε^3 \times \mathbb{R}$ for which the tangential component $T$ of the vector field $\frac{\partial}{\partial t}$ is a principal direction, where $\mathbb{Q}_ε^3$ denotes the three-dimensional non-flat Riemannian space form with sectional curvature $ε= \pm 1$, and $\frac{\partial}{\partial t}$ is the unit vector field tangent to the $\mathbb{R}$-factor. We obtain a local classification of hypersurfaces with three distinct principal curvatures satisfying specific functional relations. Then, we determine the necessary and sufficient conditions for such hypersurfaces to admit an almost Ricci soliton structure with potential vector field $T$. Finally, we prove that the only hypersurfaces admitting such solitons are rotational, by showing that the constructed examples with three distinct principal curvatures do not admit almost Ricci solitons. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_12098 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Almost Ricci Solitons on Class $\mathcal A$ Hypersurfaces of Product Spaces Çoraplı, Ahmet Umut Demirci, Burcu Bektaş Turgay, Nurettin Cenk Differential Geometry Mathematical Physics 53A10, 53C42 In this paper, we study hypersurfaces in the product spaces $\mathbb{Q}_ε^3 \times \mathbb{R}$ for which the tangential component $T$ of the vector field $\frac{\partial}{\partial t}$ is a principal direction, where $\mathbb{Q}_ε^3$ denotes the three-dimensional non-flat Riemannian space form with sectional curvature $ε= \pm 1$, and $\frac{\partial}{\partial t}$ is the unit vector field tangent to the $\mathbb{R}$-factor. We obtain a local classification of hypersurfaces with three distinct principal curvatures satisfying specific functional relations. Then, we determine the necessary and sufficient conditions for such hypersurfaces to admit an almost Ricci soliton structure with potential vector field $T$. Finally, we prove that the only hypersurfaces admitting such solitons are rotational, by showing that the constructed examples with three distinct principal curvatures do not admit almost Ricci solitons. |
| title | Almost Ricci Solitons on Class $\mathcal A$ Hypersurfaces of Product Spaces |
| topic | Differential Geometry Mathematical Physics 53A10, 53C42 |
| url | https://arxiv.org/abs/2508.12098 |