A parallel transport frame field approach to soliton surfaces associated with the Betchov–Da Rios equation in four space
UDC 514.752, 514.748 By using the parallel transport frame field, we examine the geometric properties of a soliton surface $\Psi=\Psi(s,t)$ associated with the Betchov–Da Rios equation in four-dimensional Euclidean space. We obtain derivative formulas for the parallel transport frame field of a uni...
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| Date: | 2026 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Institute of Mathematics, NAS of Ukraine
2026
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/9337 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1871646544963305472 |
|---|---|
| author | Altın, Mustafa Kazan, Ahmet Altın, Mustafa Kazan, Ahmet |
| author_facet | Altın, Mustafa Kazan, Ahmet Altın, Mustafa Kazan, Ahmet |
| author_institution_txt_mv | [
{
"author": "Mustafa Altın",
"institution": "Department of Mathematics, Faculty of Arts and Sciences, Bingöl University, Bingöl, Türkiye"
},
{
"author": "Ahmet Kazan",
"institution": "Department of Engineering Basic Sciences, Faculty of Engineering and Natural Sciences, Malatya Turgut Özal University, Malatya, Türkiye"
}
] |
| author_sort | Altın, Mustafa |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-07-24T15:24:13Z |
| description | UDC 514.752, 514.748
By using the parallel transport frame field, we examine the geometric properties of a soliton surface $\Psi=\Psi(s,t)$ associated with the Betchov–Da Rios equation in four-dimensional Euclidean space. We obtain derivative formulas for the parallel transport frame field of a unit-speed $s$-parameter curve $\Psi=\Psi(s,t),$ for all $t.$ We deduce two basic geometric invariants of the soliton surface, $k$ and $h,$ and some other important invariants, such as Gaussian curvature, mean curvature vector, and Gaussian torsion. With the aid of these, we prove a set of theorems that describe the conditions under which the soliton surface is flat, minimal, semiumbilic, or Wintgen ideal (superconformal) by using these surface invariants. In addition, we present a theorem that characterizes the curvature ellipse of the Betchov–Da Rios soliton surface with respect to the parallel transport frame field in $E^{4}.$ Finally, we construct an example of a Betchov–Da Rios soliton surface, obtain its geometric invariants, and show its projections into the three-dimensional space to illustrate our theoretical results. |
| doi_str_mv | 10.3842/umzh.v78i7-8.9337 |
| first_indexed | 2026-07-25T01:00:39Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-9337 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-25T01:00:39Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-93372026-07-24T15:24:13Z A parallel transport frame field approach to soliton surfaces associated with the Betchov–Da Rios equation in four space A parallel transport frame field approach to soliton surfaces associated with the Betchov–Da Rios equation in four space Altın, Mustafa Kazan, Ahmet Altın, Mustafa Kazan, Ahmet Betchov-Da Rios equation Parallel transport frame Curvature ellipse Wintgen inequality Differential Geometry Mathematical Physics differential equation UDC 514.752, 514.748 By using the parallel transport frame field, we examine the geometric properties of a soliton surface $\Psi=\Psi(s,t)$ associated with the Betchov–Da Rios equation in four-dimensional Euclidean space. We obtain derivative formulas for the parallel transport frame field of a unit-speed $s$-parameter curve $\Psi=\Psi(s,t),$ for all $t.$ We deduce two basic geometric invariants of the soliton surface, $k$ and $h,$ and some other important invariants, such as Gaussian curvature, mean curvature vector, and Gaussian torsion. With the aid of these, we prove a set of theorems that describe the conditions under which the soliton surface is flat, minimal, semiumbilic, or Wintgen ideal (superconformal) by using these surface invariants. In addition, we present a theorem that characterizes the curvature ellipse of the Betchov–Da Rios soliton surface with respect to the parallel transport frame field in $E^{4}.$ Finally, we construct an example of a Betchov–Da Rios soliton surface, obtain its geometric invariants, and show its projections into the three-dimensional space to illustrate our theoretical results. УДК 514.752, 514.748 Солітонні поверхні, породжені рівнянням Бетчова–Да Ріоса в чотиривимірному просторі: підхід репера паралельного перенесення Досліджено геометричні властивості солітонної поверхні $\Psi=\Psi(s,t),$ що відповідає рівнянню Бетчова–Да Ріоса в чотиривимірному евклідовому просторі, з використанням репера паралельного перенесення. Одержано формули диференціювання для репера паралельного перенесення кривої $\Psi=\Psi(s,t),$ параметризованої натуральним параметром $s,$ для всіх значень $t.$ Знайдено два основні геометричні інваріанти солітонної поверхні $k$ та $h,$ а також інші важливі інваріанти, зокрема гауссову кривину, вектор середньої кривини та гауссове кручення. За допомогою цих інваріантів доведено низку теорем, що описують умови, за яких солітонна поверхня є плоскою, мінімальною, напівумбілічною або ідеальною за Вінтгеном (суперконформною). Крім того, наведено теорему, яка характеризує еліпс кривини солітонної поверхні Бетчова–Да Ріоса щодо репера паралельного перенесення в $E^{4}.$ Насамкінець побудовано приклад солітонної поверхні Бетчова–Да Ріоса, обчислено її геометричні інваріанти та наведено її проєкції у тривимірний простір для ілюстрації одержаних теоретичних результатів. Institute of Mathematics, NAS of Ukraine 2026-07-24 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/9337 10.3842/umzh.v78i7-8.9337 Ukrains’kyi Matematychnyi Zhurnal; Vol. 78 No. 7-8 (2026); 590–591 Український математичний журнал; Том 78 № 7-8 (2026); 590–591 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/9337/10678 Copyright (c) 2026 Mustafa Altın, Ahmet Kazan |
| spellingShingle | Altın, Mustafa Kazan, Ahmet Altın, Mustafa Kazan, Ahmet A parallel transport frame field approach to soliton surfaces associated with the Betchov–Da Rios equation in four space |
| title | A parallel transport frame field approach to soliton surfaces associated with the Betchov–Da Rios equation in four space |
| title_alt | A parallel transport frame field approach to soliton surfaces associated with the Betchov–Da Rios equation in four space |
| title_full | A parallel transport frame field approach to soliton surfaces associated with the Betchov–Da Rios equation in four space |
| title_fullStr | A parallel transport frame field approach to soliton surfaces associated with the Betchov–Da Rios equation in four space |
| title_full_unstemmed | A parallel transport frame field approach to soliton surfaces associated with the Betchov–Da Rios equation in four space |
| title_short | A parallel transport frame field approach to soliton surfaces associated with the Betchov–Da Rios equation in four space |
| title_sort | parallel transport frame field approach to soliton surfaces associated with the betchov–da rios equation in four space |
| topic_facet | Betchov-Da Rios equation Parallel transport frame Curvature ellipse Wintgen inequality Differential Geometry Mathematical Physics differential equation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/9337 |
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