Flat Metrics with a Prescribed Derived Coframing
The following problem is addressed: A 3-manifold M is endowed with a triple Ω=(Ω¹, Ω², Ω³) of closed 2-forms. One wants to construct a coframing ω=(ω¹, ω², ω³) of M such that, first, dωi=Ωi for i=1,2,3, and, second, the Riemannian metric g = (ω¹)² +(ω²)²+(ω³)² be flat. We show that, in the 'n...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2020 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2020
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/210606 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Flat Metrics with a Prescribed Derived Coframing. Robert L. Bryant and Jeanne N. Clelland. SIGMA 16 (2020), 004, 23 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862542429539794944 |
|---|---|
| author | Bryant, Robert L. Clelland, Jeanne N. |
| author_facet | Bryant, Robert L. Clelland, Jeanne N. |
| citation_txt | Flat Metrics with a Prescribed Derived Coframing. Robert L. Bryant and Jeanne N. Clelland. SIGMA 16 (2020), 004, 23 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The following problem is addressed: A 3-manifold M is endowed with a triple Ω=(Ω¹, Ω², Ω³) of closed 2-forms. One wants to construct a coframing ω=(ω¹, ω², ω³) of M such that, first, dωi=Ωi for i=1,2,3, and, second, the Riemannian metric g = (ω¹)² +(ω²)²+(ω³)² be flat. We show that, in the 'nonsingular case', i.e., when the three 2-forms Ωⁱₚ span at least a 2-dimensional subspace of Λ²(T*ₚM) and are real-analytic in some p-centered coordinates, this problem is always solvable on a neighborhood of p∈M, with the general solution ω depending on three arbitrary functions of two variables. Moreover, the characteristic variety of the generic solution ω can be taken to be a nonsingular cubic. Some singular situations are considered as well. In particular, we show that the problem is solvable locally when Ω¹, Ω², Ω³ are scalar multiples of a single 2-form that do not vanish simultaneously and satisfy a nondegeneracy condition. We also show by example that solutions may fail to exist when these conditions are not satisfied.
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| first_indexed | 2025-12-17T12:03:35Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-210606 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:03:35Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bryant, Robert L. Clelland, Jeanne N. 2025-12-12T10:40:12Z 2020 Flat Metrics with a Prescribed Derived Coframing. Robert L. Bryant and Jeanne N. Clelland. SIGMA 16 (2020), 004, 23 pages 1815-0659 2010 Mathematics Subject Classification: 53A55; 53B15 arXiv:1908.01041 https://nasplib.isofts.kiev.ua/handle/123456789/210606 https://doi.org/10.3842/SIGMA.2020.004 The following problem is addressed: A 3-manifold M is endowed with a triple Ω=(Ω¹, Ω², Ω³) of closed 2-forms. One wants to construct a coframing ω=(ω¹, ω², ω³) of M such that, first, dωi=Ωi for i=1,2,3, and, second, the Riemannian metric g = (ω¹)² +(ω²)²+(ω³)² be flat. We show that, in the 'nonsingular case', i.e., when the three 2-forms Ωⁱₚ span at least a 2-dimensional subspace of Λ²(T*ₚM) and are real-analytic in some p-centered coordinates, this problem is always solvable on a neighborhood of p∈M, with the general solution ω depending on three arbitrary functions of two variables. Moreover, the characteristic variety of the generic solution ω can be taken to be a nonsingular cubic. Some singular situations are considered as well. In particular, we show that the problem is solvable locally when Ω¹, Ω², Ω³ are scalar multiples of a single 2-form that do not vanish simultaneously and satisfy a nondegeneracy condition. We also show by example that solutions may fail to exist when these conditions are not satisfied. Thanks to Duke University for its support via a research grant (Bryant), to the National Science Foundation for its support via research grant DMS-1206272 (Clelland), and to the Simons Foundation for its support via a Collaboration Grant for Mathematicians (Clelland). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Flat Metrics with a Prescribed Derived Coframing Article published earlier |
| spellingShingle | Flat Metrics with a Prescribed Derived Coframing Bryant, Robert L. Clelland, Jeanne N. |
| title | Flat Metrics with a Prescribed Derived Coframing |
| title_full | Flat Metrics with a Prescribed Derived Coframing |
| title_fullStr | Flat Metrics with a Prescribed Derived Coframing |
| title_full_unstemmed | Flat Metrics with a Prescribed Derived Coframing |
| title_short | Flat Metrics with a Prescribed Derived Coframing |
| title_sort | flat metrics with a prescribed derived coframing |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210606 |
| work_keys_str_mv | AT bryantrobertl flatmetricswithaprescribedderivedcoframing AT clellandjeannen flatmetricswithaprescribedderivedcoframing |