Twisted-Austere Submanifolds in Euclidean Space
A twisted-austere -fold (, ) in ℝⁿ consists of a -dimensional submanifold of ℝⁿ together with a closed 1-form on , such that the second fundamental form A of and the 1-form satisfy a particular system of coupled nonlinear second-order PDE. Given such an object, the ''twisted conormal b...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2021 |
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2021
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211165 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Twisted-Austere Submanifolds in Euclidean Space. Thomas A. Ivey and Spiro Karigiannis. SIGMA 17 (2021), 023, 31 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862652930405957632 |
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| author | Ivey, Thomas A. Karigiannis, Spiro |
| author_facet | Ivey, Thomas A. Karigiannis, Spiro |
| citation_txt | Twisted-Austere Submanifolds in Euclidean Space. Thomas A. Ivey and Spiro Karigiannis. SIGMA 17 (2021), 023, 31 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | A twisted-austere -fold (, ) in ℝⁿ consists of a -dimensional submanifold of ℝⁿ together with a closed 1-form on , such that the second fundamental form A of and the 1-form satisfy a particular system of coupled nonlinear second-order PDE. Given such an object, the ''twisted conormal bundle'' *+ is a special Lagrangian submanifold of ℂⁿ. We review the twisted austere condition and give an explicit example. Then we focus on twisted austere 3-folds. We give a geometric description of all solutions when the ''base'' is a cylinder, and when is austere. Finally, we prove that, other than the case of a generalized helicoid in ℝ⁵ discovered by Bryant, there are no other possibilities for the base . This gives a complete classification of twisted-austere 3-folds in Rⁿ.
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| first_indexed | 2026-03-15T16:57:02Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211165 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T16:57:02Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ivey, Thomas A. Karigiannis, Spiro 2025-12-25T13:20:08Z 2021 Twisted-Austere Submanifolds in Euclidean Space. Thomas A. Ivey and Spiro Karigiannis. SIGMA 17 (2021), 023, 31 pages 1815-0659 2020 Mathematics Subject Classification: 53B25; 53C38; 53C40; 53D12; 58A15 arXiv:2006.15119 https://nasplib.isofts.kiev.ua/handle/123456789/211165 https://doi.org/10.3842/SIGMA.2021.023 A twisted-austere -fold (, ) in ℝⁿ consists of a -dimensional submanifold of ℝⁿ together with a closed 1-form on , such that the second fundamental form A of and the 1-form satisfy a particular system of coupled nonlinear second-order PDE. Given such an object, the ''twisted conormal bundle'' *+ is a special Lagrangian submanifold of ℂⁿ. We review the twisted austere condition and give an explicit example. Then we focus on twisted austere 3-folds. We give a geometric description of all solutions when the ''base'' is a cylinder, and when is austere. Finally, we prove that, other than the case of a generalized helicoid in ℝ⁵ discovered by Bryant, there are no other possibilities for the base . This gives a complete classification of twisted-austere 3-folds in Rⁿ. The authors thank the anonymous referees for useful feedback and comments that improved the quality of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Twisted-Austere Submanifolds in Euclidean Space Article published earlier |
| spellingShingle | Twisted-Austere Submanifolds in Euclidean Space Ivey, Thomas A. Karigiannis, Spiro |
| title | Twisted-Austere Submanifolds in Euclidean Space |
| title_full | Twisted-Austere Submanifolds in Euclidean Space |
| title_fullStr | Twisted-Austere Submanifolds in Euclidean Space |
| title_full_unstemmed | Twisted-Austere Submanifolds in Euclidean Space |
| title_short | Twisted-Austere Submanifolds in Euclidean Space |
| title_sort | twisted-austere submanifolds in euclidean space |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211165 |
| work_keys_str_mv | AT iveythomasa twistedausteresubmanifoldsineuclideanspace AT karigiannisspiro twistedausteresubmanifoldsineuclideanspace |