The Geometry of Almost Einstein (2,3,5) Distributions
We analyze the classic problem of existence of Einstein metrics in a given conformal structure for the class of conformal structures inducedf Nurowski's construction by (oriented) (2,3,5) distributions. We characterize in two ways such conformal structures that admit an almost Einstein scale: F...
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| Дата: | 2017 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2017
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/148555 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | The Geometry of Almost Einstein (2,3,5) Distributions / K. Sagerschnig, T. Willse // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 67 назв. — англ. |
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irk-123456789-1485552019-02-19T01:26:07Z The Geometry of Almost Einstein (2,3,5) Distributions Sagerschnig, K. Willse, T. We analyze the classic problem of existence of Einstein metrics in a given conformal structure for the class of conformal structures inducedf Nurowski's construction by (oriented) (2,3,5) distributions. We characterize in two ways such conformal structures that admit an almost Einstein scale: First, they are precisely the oriented conformal structures c that are induced by at least two distinct oriented (2,3,5) distributions; in this case there is a 1-parameter family of such distributions that induce c. Second, they are characterized by the existence of a holonomy reduction to SU(1,2), SL(3,R), or a particular semidirect product SL(2,R)⋉Q+, according to the sign of the Einstein constant of the corresponding metric. Via the curved orbit decomposition formalism such a reduction partitions the underlying manifold into several submanifolds and endows each ith a geometric structure. This establishes novel links between (2,3,5) distributions and many other geometries - several classical geometries among them - including: Sasaki-Einstein geometry and its paracomplex and null-complex analogues in dimension 5; Kähler-Einstein geometry and its paracomplex and null-complex analogues, Fefferman Lorentzian conformal structures, and para-Fefferman neutral conformal structures in dimension 4; CR geometry and the point geometry of second-order ordinary differential equations in dimension 3; and projective geometry in dimension 2. We describe a generalized Fefferman construction that builds from a 4-dimensional Kähler-Einstein or para-Kähler-Einstein structure a family of (2,3,5) distributions that induce the same (Einstein) conformal structure. We exploit some of these links to construct new examples, establishing the existence of nonflat almost Einstein (2,3,5) conformal structures for which the Einstein constant is positive and negative. 2017 Article The Geometry of Almost Einstein (2,3,5) Distributions / K. Sagerschnig, T. Willse // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 67 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 32Q20; 32V05; 53A30; 53A40; 53B35; 53C15; 53C25; 53C29; 53C55; 58A30 DOI:10.3842/SIGMA.2017.004 http://dspace.nbuv.gov.ua/handle/123456789/148555 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We analyze the classic problem of existence of Einstein metrics in a given conformal structure for the class of conformal structures inducedf Nurowski's construction by (oriented) (2,3,5) distributions. We characterize in two ways such conformal structures that admit an almost Einstein scale: First, they are precisely the oriented conformal structures c that are induced by at least two distinct oriented (2,3,5) distributions; in this case there is a 1-parameter family of such distributions that induce c. Second, they are characterized by the existence of a holonomy reduction to SU(1,2), SL(3,R), or a particular semidirect product SL(2,R)⋉Q+, according to the sign of the Einstein constant of the corresponding metric. Via the curved orbit decomposition formalism such a reduction partitions the underlying manifold into several submanifolds and endows each ith a geometric structure. This establishes novel links between (2,3,5) distributions and many other geometries - several classical geometries among them - including: Sasaki-Einstein geometry and its paracomplex and null-complex analogues in dimension 5; Kähler-Einstein geometry and its paracomplex and null-complex analogues, Fefferman Lorentzian conformal structures, and para-Fefferman neutral conformal structures in dimension 4; CR geometry and the point geometry of second-order ordinary differential equations in dimension 3; and projective geometry in dimension 2. We describe a generalized Fefferman construction that builds from a 4-dimensional Kähler-Einstein or para-Kähler-Einstein structure a family of (2,3,5) distributions that induce the same (Einstein) conformal structure. We exploit some of these links to construct new examples, establishing the existence of nonflat almost Einstein (2,3,5) conformal structures for which the Einstein constant is positive and negative. |
| format |
Article |
| author |
Sagerschnig, K. Willse, T. |
| spellingShingle |
Sagerschnig, K. Willse, T. The Geometry of Almost Einstein (2,3,5) Distributions Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Sagerschnig, K. Willse, T. |
| author_sort |
Sagerschnig, K. |
| title |
The Geometry of Almost Einstein (2,3,5) Distributions |
| title_short |
The Geometry of Almost Einstein (2,3,5) Distributions |
| title_full |
The Geometry of Almost Einstein (2,3,5) Distributions |
| title_fullStr |
The Geometry of Almost Einstein (2,3,5) Distributions |
| title_full_unstemmed |
The Geometry of Almost Einstein (2,3,5) Distributions |
| title_sort |
geometry of almost einstein (2,3,5) distributions |
| publisher |
Інститут математики НАН України |
| publishDate |
2017 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/148555 |
| citation_txt |
The Geometry of Almost Einstein (2,3,5) Distributions / K. Sagerschnig, T. Willse // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 67 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2023-05-20T17:29:44Z |
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2023-05-20T17:29:44Z |
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1796153420765200384 |