A Geometric Application of Soliton Surfaces associated with the Betchov-Da Rios Equation using an Extended Darboux Frame Field in $E^{4}$
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| Format: | Preprint |
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2024
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| _version_ | 1866911910503383040 |
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| author | Kazan, Ahmet Altın, Mustafa |
| author_facet | Kazan, Ahmet Altın, Mustafa |
| contents | In this paper, for a soliton surface $Ω=Ω(u,v)$ associated with the Betchov-Da Rios equation, we obtain the derivative formulas of an extended Darboux frame field of a unit speed curve $u$-parameter curve $Ω=Ω(u,v)$ for all $v$. Also, we get the geometric invariants $k$ and $h$ of the soliton surface $Ω=Ω(u,v)$ and we obtain the Gaussian curvature, mean curvature vector and Gaussian torsion of $Ω$. We give some important geometric characterizations such as flatness, minimality and semi-umbilicaly with the aid of these invariants. Additionally, we study the curvature ellipse of the Betchov-Da Rios soliton surface and Wintgen ideal (superconformal) Betchov-Da Rios soliton surface with respect to an extended Darboux frame field. Finally, we construct an application for the Betchov-Da Rios soliton surface with the aid of an extended Darboux frame field. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_05139 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Geometric Application of Soliton Surfaces associated with the Betchov-Da Rios Equation using an Extended Darboux Frame Field in $E^{4}$ Kazan, Ahmet Altın, Mustafa Differential Geometry In this paper, for a soliton surface $Ω=Ω(u,v)$ associated with the Betchov-Da Rios equation, we obtain the derivative formulas of an extended Darboux frame field of a unit speed curve $u$-parameter curve $Ω=Ω(u,v)$ for all $v$. Also, we get the geometric invariants $k$ and $h$ of the soliton surface $Ω=Ω(u,v)$ and we obtain the Gaussian curvature, mean curvature vector and Gaussian torsion of $Ω$. We give some important geometric characterizations such as flatness, minimality and semi-umbilicaly with the aid of these invariants. Additionally, we study the curvature ellipse of the Betchov-Da Rios soliton surface and Wintgen ideal (superconformal) Betchov-Da Rios soliton surface with respect to an extended Darboux frame field. Finally, we construct an application for the Betchov-Da Rios soliton surface with the aid of an extended Darboux frame field. |
| title | A Geometric Application of Soliton Surfaces associated with the Betchov-Da Rios Equation using an Extended Darboux Frame Field in $E^{4}$ |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2406.05139 |