Bertrand and hyperelastic curves on Riemannian manifolds
UDC 514.76, 514.75 We introduce the notion of Bertrand curves on Riemannian manifolds. The necessary conditions for a curve on a Riemannian manifold to be a Bertrand curve are obtained in the form of differential equations. Then, necessary and sufficient conditions are found for the Bertrand curve t...
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| Datum: | 2026 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Institute of Mathematics, NAS of Ukraine
2026
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/8357 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1866663706579959808 |
|---|---|
| author | Çalışkan, Abdussamet Çalışkan, Abdussamet |
| author_facet | Çalışkan, Abdussamet Çalışkan, Abdussamet |
| author_institution_txt_mv | [
{
"author": "Abdussamet Çalışkan",
"institution": "Fatsa Vocational School, Accounting and Tax Applications Ordu University, Ordu, Turkey"
}
] |
| author_sort | Çalışkan, Abdussamet |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-05-30T12:43:58Z |
| description | UDC 514.76, 514.75
We introduce the notion of Bertrand curves on Riemannian manifolds. The necessary conditions for a curve on a Riemannian manifold to be a Bertrand curve are obtained in the form of differential equations. Then, necessary and sufficient conditions are found for the Bertrand curve to remain a Bertrand curve in the ambient space. Moreover, we examine the influence of Bertrand and hyperelastic curves on the geometric properties of isometric immersions. We study the relationship between these types of curves and umbilical submanifolds. Finally, we present the criteria for the Bertrand curve to be a hyperelastic curve in the ambient space along an immersion. |
| doi_str_mv | 10.3842/umzh.v78i5-6.8357 |
| first_indexed | 2026-05-30T01:00:36Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-8357 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-05-31T01:00:34Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-83572026-05-30T12:43:58Z Bertrand and hyperelastic curves on Riemannian manifolds Bertrand and hyperelastic curves on Riemannian manifolds Çalışkan, Abdussamet Çalışkan, Abdussamet Bertrand curve hyperelastic curve Riemannian manifold umbilical, totally geodesic submanifold UDC 514.76, 514.75 We introduce the notion of Bertrand curves on Riemannian manifolds. The necessary conditions for a curve on a Riemannian manifold to be a Bertrand curve are obtained in the form of differential equations. Then, necessary and sufficient conditions are found for the Bertrand curve to remain a Bertrand curve in the ambient space. Moreover, we examine the influence of Bertrand and hyperelastic curves on the geometric properties of isometric immersions. We study the relationship between these types of curves and umbilical submanifolds. Finally, we present the criteria for the Bertrand curve to be a hyperelastic curve in the ambient space along an immersion. УДК 514.76, 514.75 Криві Бертрана та гіпереластичні криві на ріманових многовидах Введено поняття кривих Бертрана на ріманових многовидах. Необхідні умови для того, щоб крива на рімановому многовиді була кривою Бертрана, отримано у вигляді диференціальних рівнянь. Знайдено необхідні та достатні умови, за яких крива Бертрана залишається кривою Бертрана в навколишньому просторі вздовж занурення. Досліджено вплив кривих Бертрана та гіпереластичних кривих на геометричні властивості ізометричних занурень. Вивчено зв'язок між цими типами кривих та омбілічними підмноговидами. Крім того, наведено критерії того, що крива Бертрана буде гіпереластичною кривою в навколишньому просторі вздовж занурення. Institute of Mathematics, NAS of Ukraine 2026-05-29 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/8357 10.3842/umzh.v78i5-6.8357 Ukrains’kyi Matematychnyi Zhurnal; Vol. 78 No. 5-6 (2026); 363 Український математичний журнал; Том 78 № 5-6 (2026); 363 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/8357/10654 Copyright (c) 2026 Abdussamet Çalışkan |
| spellingShingle | Çalışkan, Abdussamet Çalışkan, Abdussamet Bertrand and hyperelastic curves on Riemannian manifolds |
| title | Bertrand and hyperelastic curves on Riemannian manifolds |
| title_alt | Bertrand and hyperelastic curves on Riemannian manifolds |
| title_full | Bertrand and hyperelastic curves on Riemannian manifolds |
| title_fullStr | Bertrand and hyperelastic curves on Riemannian manifolds |
| title_full_unstemmed | Bertrand and hyperelastic curves on Riemannian manifolds |
| title_short | Bertrand and hyperelastic curves on Riemannian manifolds |
| title_sort | bertrand and hyperelastic curves on riemannian manifolds |
| topic_facet | Bertrand curve hyperelastic curve Riemannian manifold umbilical, totally geodesic submanifold |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/8357 |
| work_keys_str_mv | AT calıskanabdussamet bertrandandhyperelasticcurvesonriemannianmanifolds AT calıskanabdussamet bertrandandhyperelasticcurvesonriemannianmanifolds |