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...
Збережено в:
| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2020 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2020
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/210606 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| 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| id |
nasplib_isofts_kiev_ua-123456789-210606 |
|---|---|
| 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 |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Flat Metrics with a Prescribed Derived Coframing |
| spellingShingle |
Flat Metrics with a Prescribed Derived Coframing Bryant, Robert L. Clelland, Jeanne N. |
| title_short |
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_sort |
flat metrics with a prescribed derived coframing |
| author |
Bryant, Robert L. Clelland, Jeanne N. |
| author_facet |
Bryant, Robert L. Clelland, Jeanne N. |
| publishDate |
2020 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| 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.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210606 |
| citation_txt |
Flat Metrics with a Prescribed Derived Coframing. Robert L. Bryant and Jeanne N. Clelland. SIGMA 16 (2020), 004, 23 pages |
| work_keys_str_mv |
AT bryantrobertl flatmetricswithaprescribedderivedcoframing AT clellandjeannen flatmetricswithaprescribedderivedcoframing |
| first_indexed |
2025-12-17T12:03:35Z |
| last_indexed |
2025-12-17T12:03:35Z |
| _version_ |
1851756920238505985 |