On space-like constant slope surfaces and Bertrand curves in Minkowski 3-space
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2011
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| _version_ | 1866918349776093184 |
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| author | Babaarslan, Murat Yayli, Yusuf |
| author_facet | Babaarslan, Murat Yayli, Yusuf |
| contents | In the present paper, we define the notions of Lorentzian Sabban frames and de Sitter evolutes of the unit speed space-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$. In addition, we investigate the invariants and geometric properties of these curves. Afterwards, we show that space-like Bertrand curves and time-like Bertrand curves can be constructed from unit speed space-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$ and hyperbolic space $\mathbb{H}^{2}$, respectively. We obtain the relations between Bertrand curves and helices. Also we show that pseudo-spherical Darboux images of Bertrand curves are equal to pseudo-spherical evolutes in Minkowski 3-space $\mathbb{R}^{3}_{1}$. Moreover we investigate the relations between Bertrand curves and space-like constant slope surfaces in $\mathbb{R}^{3}_{1}$. Finally, we give some examples to illustrate our main results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1112_1504 |
| institution | arXiv |
| publishDate | 2011 |
| record_format | arxiv |
| spellingShingle | On space-like constant slope surfaces and Bertrand curves in Minkowski 3-space Babaarslan, Murat Yayli, Yusuf Differential Geometry Mathematical Physics 53B25 In the present paper, we define the notions of Lorentzian Sabban frames and de Sitter evolutes of the unit speed space-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$. In addition, we investigate the invariants and geometric properties of these curves. Afterwards, we show that space-like Bertrand curves and time-like Bertrand curves can be constructed from unit speed space-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$ and hyperbolic space $\mathbb{H}^{2}$, respectively. We obtain the relations between Bertrand curves and helices. Also we show that pseudo-spherical Darboux images of Bertrand curves are equal to pseudo-spherical evolutes in Minkowski 3-space $\mathbb{R}^{3}_{1}$. Moreover we investigate the relations between Bertrand curves and space-like constant slope surfaces in $\mathbb{R}^{3}_{1}$. Finally, we give some examples to illustrate our main results. |
| title | On space-like constant slope surfaces and Bertrand curves in Minkowski 3-space |
| topic | Differential Geometry Mathematical Physics 53B25 |
| url | https://arxiv.org/abs/1112.1504 |