Generalized Bishop frames of regular time-like curves in 4-dimensional Lorentz space $\mathbb{L}^{4}$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912640914161664 |
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| author | Nomoto, Subaru |
| author_facet | Nomoto, Subaru |
| contents | We introduced generalized Bishop frames on curves in 4-dimensional Euclidean space $\mathbb{E}^{4}$, which are orthonormal frames such that the derivatives of the vectors of the frames along the curve can be expressed, via a certain matrix, as a linear combination of the vectors of the frame.
In relation to that, we study generalized Bishop frames of regular time-like curves. In a previous work, we showed that there is a hierarchy among different types of generalized Bishop frames for regular curves in the Euclidean space. Building upon this study, we further investigate it in the 4-dimensional Lorentz space $\mathbb{L}^4$. There are four types of generalized Bishop frames of regular time-like curves in $\mathbb{L}^{4}$ up to the change of the order of vectors fixing the first one which is the tangent vector. Unlike other types of curves, such as light-like and space-like ones, the time-like curve can be investigated in a manner analogous to the Euclidean case. We find that a hierarchy of frames exists, similar to that in the Euclidean setting. Based on this hierarchy, we propose a new classification of curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_09345 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Bishop frames of regular time-like curves in 4-dimensional Lorentz space $\mathbb{L}^{4}$ Nomoto, Subaru Differential Geometry 53A04, \, 53B30, \, 53C42, \, 53A45 We introduced generalized Bishop frames on curves in 4-dimensional Euclidean space $\mathbb{E}^{4}$, which are orthonormal frames such that the derivatives of the vectors of the frames along the curve can be expressed, via a certain matrix, as a linear combination of the vectors of the frame. In relation to that, we study generalized Bishop frames of regular time-like curves. In a previous work, we showed that there is a hierarchy among different types of generalized Bishop frames for regular curves in the Euclidean space. Building upon this study, we further investigate it in the 4-dimensional Lorentz space $\mathbb{L}^4$. There are four types of generalized Bishop frames of regular time-like curves in $\mathbb{L}^{4}$ up to the change of the order of vectors fixing the first one which is the tangent vector. Unlike other types of curves, such as light-like and space-like ones, the time-like curve can be investigated in a manner analogous to the Euclidean case. We find that a hierarchy of frames exists, similar to that in the Euclidean setting. Based on this hierarchy, we propose a new classification of curves. |
| title | Generalized Bishop frames of regular time-like curves in 4-dimensional Lorentz space $\mathbb{L}^{4}$ |
| topic | Differential Geometry 53A04, \, 53B30, \, 53C42, \, 53A45 |
| url | https://arxiv.org/abs/2510.09345 |