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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| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
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| Zusammenfassung: | 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. |
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| DOI: | 10.3842/umzh.v78i5-6.8357 |