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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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2021 |
| ISSN: | 1815-0659 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2021
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211165 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | 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| Summary: | 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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| ISSN: | 1815-0659 |