Conformal trajectories in 3-dimensional space form
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909210343636992 |
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| author | Lopez, Rafael Munteanu, Marian Ioan |
| author_facet | Lopez, Rafael Munteanu, Marian Ioan |
| contents | We introduce the notion of conformal trajectories in three-dimensional Riemannian manifolds $M^3$. Given a conformal vector field $V\in\mathfrak{X}(M^3)$, a conformal trajectory of $V$ is a regular curve $γ$ in $M^3$ satisfying $\nabla_{γ'}γ'=q\, V\timesγ'$, for some fixed non-zero constant $q\in {\mathbb{R}}$. In this paper, we study conformal trajectories in the space forms ${\mathbb{R}}^3$, ${\mathbb{S}}^3$ and ${\mathbb{H}}^3$. For (non-Killing) conformal vector fields in ${\mathbb{S}}^3$ (respectively in ${\mathbb{H}}^3$), we prove that conformal trajectories have constant curvature and its torsion is a linear combination of trigonometric (respectively hyperbolic) functions on the arc-length parameter. In the case of Euclidean space ${\mathbb{R}}^3$, we obtain the same result for the radial vector field and characterising all conformal trajectories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_15890 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conformal trajectories in 3-dimensional space form Lopez, Rafael Munteanu, Marian Ioan Differential Geometry Primary 53A10, Secondary 53C44, 53C21, 53C42 We introduce the notion of conformal trajectories in three-dimensional Riemannian manifolds $M^3$. Given a conformal vector field $V\in\mathfrak{X}(M^3)$, a conformal trajectory of $V$ is a regular curve $γ$ in $M^3$ satisfying $\nabla_{γ'}γ'=q\, V\timesγ'$, for some fixed non-zero constant $q\in {\mathbb{R}}$. In this paper, we study conformal trajectories in the space forms ${\mathbb{R}}^3$, ${\mathbb{S}}^3$ and ${\mathbb{H}}^3$. For (non-Killing) conformal vector fields in ${\mathbb{S}}^3$ (respectively in ${\mathbb{H}}^3$), we prove that conformal trajectories have constant curvature and its torsion is a linear combination of trigonometric (respectively hyperbolic) functions on the arc-length parameter. In the case of Euclidean space ${\mathbb{R}}^3$, we obtain the same result for the radial vector field and characterising all conformal trajectories. |
| title | Conformal trajectories in 3-dimensional space form |
| topic | Differential Geometry Primary 53A10, Secondary 53C44, 53C21, 53C42 |
| url | https://arxiv.org/abs/2405.15890 |