A Parallel Transport Frame Field Approach to Soliton Surfaces associated with the Betchov-Da Rios Equation in Four Space
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913932671713280 |
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| author | Altin, Mustafa Kazan, Ahmet |
| author_facet | Altin, Mustafa Kazan, Ahmet |
| contents | In the present paper, the geometric properties of a soliton surface $Ψ=Ψ(s,t)$ associated with the Betchov-Da Rios (B-DR) equation using the parallel transport frame field in four-dimensional Euclidean space are examined. We obtain the derivative formulas for the parallel transport frame field of a unit-speed $s$-parameter curve $Ψ=Ψ(s,t),$ for all $t$. We obtain the soliton surface's two basic geometric invariants, $k$ and $h,$ and some other important invariants such as Gaussian curvature, mean curvature vector and Gaussian torsion. With the aid of these, a set of theorems describing the conditions in which the soliton surface is flat, minimal, semi-umbilic or Wintgen ideal (superconformal) are proved using these surface invariants. Also, we give a theorem which characterizes the curvature ellipse of the B-DR soliton surface with respect to the parallel transport frame field in $E^{4}$. Finally, we construct an example of a B-DR soliton surface, obtain its geometric invariants and showed its projections into three-dimensional space to demonstrate our theoretical understanding. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06248 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Parallel Transport Frame Field Approach to Soliton Surfaces associated with the Betchov-Da Rios Equation in Four Space Altin, Mustafa Kazan, Ahmet Differential Geometry In the present paper, the geometric properties of a soliton surface $Ψ=Ψ(s,t)$ associated with the Betchov-Da Rios (B-DR) equation using the parallel transport frame field in four-dimensional Euclidean space are examined. We obtain the derivative formulas for the parallel transport frame field of a unit-speed $s$-parameter curve $Ψ=Ψ(s,t),$ for all $t$. We obtain the soliton surface's two basic geometric invariants, $k$ and $h,$ and some other important invariants such as Gaussian curvature, mean curvature vector and Gaussian torsion. With the aid of these, a set of theorems describing the conditions in which the soliton surface is flat, minimal, semi-umbilic or Wintgen ideal (superconformal) are proved using these surface invariants. Also, we give a theorem which characterizes the curvature ellipse of the B-DR soliton surface with respect to the parallel transport frame field in $E^{4}$. Finally, we construct an example of a B-DR soliton surface, obtain its geometric invariants and showed its projections into three-dimensional space to demonstrate our theoretical understanding. |
| title | A Parallel Transport Frame Field Approach to Soliton Surfaces associated with the Betchov-Da Rios Equation in Four Space |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2507.06248 |