On Integrable Nets in General and Concordant Chebyshev Nets in Particular
We consider general integrable curve nets in Euclidean space as a particular integrable geometry invariant with respect to rigid motions and net-preserving reparameterisations. For their description, we first give an overview of the most important second-order invariants and relations among them. As...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2025 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2025
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/213185 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On Integrable Nets in General and Concordant Chebyshev Nets in Particular. Michal Marvan. SIGMA 21 (2025), 029, 34 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862546578005295104 |
|---|---|
| author | Marvan, Michal |
| author_facet | Marvan, Michal |
| citation_txt | On Integrable Nets in General and Concordant Chebyshev Nets in Particular. Michal Marvan. SIGMA 21 (2025), 029, 34 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We consider general integrable curve nets in Euclidean space as a particular integrable geometry invariant with respect to rigid motions and net-preserving reparameterisations. For their description, we first give an overview of the most important second-order invariants and relations among them. As a particular integrable example, we reinterpret the result of I.S. Krasil'shchik and M. Marvan (see Section 2, Case 2 in [Acta Appl. Math. 56 (1999), 217-230]) as a curve net satisfying an ℝ-linear relation between the Schief curvature of the net and the Gauss curvature of the supporting surface. In the special case when the curvatures are proportional (concordant nets), we find a correspondence to pairs of pseudospherical surfaces of equal negative constant Gaussian curvatures. Conversely, we also show that two generic pseudospherical surfaces of equal negative constant Gaussian curvatures induce a concordant Chebyshev net. The construction generalises the well-known correspondence between pairs of curves and translation surfaces.
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| first_indexed | 2026-03-21T11:41:17Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-213185 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T11:41:17Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Marvan, Michal 2026-02-16T16:35:07Z 2025 On Integrable Nets in General and Concordant Chebyshev Nets in Particular. Michal Marvan. SIGMA 21 (2025), 029, 34 pages 1815-0659 2020 Mathematics Subject Classification: 37K10; 53A05; 53A55; 53A60 arXiv:2403.12626 https://nasplib.isofts.kiev.ua/handle/123456789/213185 https://doi.org/10.3842/SIGMA.2025.029 We consider general integrable curve nets in Euclidean space as a particular integrable geometry invariant with respect to rigid motions and net-preserving reparameterisations. For their description, we first give an overview of the most important second-order invariants and relations among them. As a particular integrable example, we reinterpret the result of I.S. Krasil'shchik and M. Marvan (see Section 2, Case 2 in [Acta Appl. Math. 56 (1999), 217-230]) as a curve net satisfying an ℝ-linear relation between the Schief curvature of the net and the Gauss curvature of the supporting surface. In the special case when the curvatures are proportional (concordant nets), we find a correspondence to pairs of pseudospherical surfaces of equal negative constant Gaussian curvatures. Conversely, we also show that two generic pseudospherical surfaces of equal negative constant Gaussian curvatures induce a concordant Chebyshev net. The construction generalises the well-known correspondence between pairs of curves and translation surfaces. This research received support from MŠMT under RVO 47813059. The author is grateful to Evgeny Ferapontov and Jan Cieśliński for the introduction to integrable surfaces and thought-provoking discussions that inspired this particular research. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Integrable Nets in General and Concordant Chebyshev Nets in Particular Article published earlier |
| spellingShingle | On Integrable Nets in General and Concordant Chebyshev Nets in Particular Marvan, Michal |
| title | On Integrable Nets in General and Concordant Chebyshev Nets in Particular |
| title_full | On Integrable Nets in General and Concordant Chebyshev Nets in Particular |
| title_fullStr | On Integrable Nets in General and Concordant Chebyshev Nets in Particular |
| title_full_unstemmed | On Integrable Nets in General and Concordant Chebyshev Nets in Particular |
| title_short | On Integrable Nets in General and Concordant Chebyshev Nets in Particular |
| title_sort | on integrable nets in general and concordant chebyshev nets in particular |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213185 |
| work_keys_str_mv | AT marvanmichal onintegrablenetsingeneralandconcordantchebyshevnetsinparticular |