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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2510.10247 |
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| _version_ | 1866916031435374592 |
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| author | Pinteaux, Constant Tuynman, Gijs M. |
| author_facet | Pinteaux, Constant Tuynman, Gijs M. |
| contents | Given a submanifold $M\subset \mathbf{R}^ν$, a curve $γ:I\to M$ and tangent vectors $v$ along $γ$, we roll the tangent space along $γ$. In doing so, we get an imprint/trace of $γ$ on the tangent space, as well as an imprint/trace of the tangent vectors. We show that for a vector field $v$ along $γ$, the imprint/trace of its covariant derivative is the ordinary derivative of its imprint/trace vector field. It then follows easily that $v$ is a set of parallel vectors along $γ$ if and only if their imprint/trace on the (affine) tangent space is constant and that $γ$ is a geodesic if and only if its trace on the tangent space is a straight line. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10247 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The rolling tangent space, a forgotten vision on parallel transport and geodesics Pinteaux, Constant Tuynman, Gijs M. Differential Geometry 53A99 Given a submanifold $M\subset \mathbf{R}^ν$, a curve $γ:I\to M$ and tangent vectors $v$ along $γ$, we roll the tangent space along $γ$. In doing so, we get an imprint/trace of $γ$ on the tangent space, as well as an imprint/trace of the tangent vectors. We show that for a vector field $v$ along $γ$, the imprint/trace of its covariant derivative is the ordinary derivative of its imprint/trace vector field. It then follows easily that $v$ is a set of parallel vectors along $γ$ if and only if their imprint/trace on the (affine) tangent space is constant and that $γ$ is a geodesic if and only if its trace on the tangent space is a straight line. |
| title | The rolling tangent space, a forgotten vision on parallel transport and geodesics |
| topic | Differential Geometry 53A99 |
| url | https://arxiv.org/abs/2510.10247 |