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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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2017 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2017
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| Онлайн доступ: | https://nasplib.isofts.kiev.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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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862720125187129344 |
|---|---|
| author | Sagerschnig, K. Willse, T. |
| author_facet | Sagerschnig, K. Willse, T. |
| 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| 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.
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| first_indexed | 2025-12-07T18:23:46Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-148555 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:23:46Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Sagerschnig, K. Willse, T. 2019-02-18T15:32:12Z 2019-02-18T15:32:12Z 2017 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 https://nasplib.isofts.kiev.ua/handle/123456789/148555 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. It is a pleasure to thank Andreas Cap for discussions about curved orbit decompositions and ˇ
 natural operators on 3-dimensional CR and Legendrean contact structures, Boris Doubrov and
 Boris Kruglikov for discussions about the geometry of second-order ODEs modulo point transformations,
 Rod Gover for comments about conformal tractor geometry, John Huerta for comments
 about the algebra of G2, Pawe l Nurowski for a suggestion that gave rise to Example 6.1,
 and Michael Eastwood and Dennis The for comments about various aspects of the project.
 Ian Anderson’s Maple package DifferentialGeometry was used extensively, including for the
 derivation of Proposition 4.1 and Algorithm 4.12 and the preparation of Example 6.2, and it
 is again a pleasure to thank him for helpful comments about the package’s usage. Finally, the
 authors thank the referees for several helpful comments and suggestions.
 The first author is an INdAM (Istituto Nazionale di Alta Matematica) research fellow. She
 gratefully acknowledges support from the Austrian Science Fund (FWF) via project J3071–N13
 and support from project FIR–2013 Geometria delle equazioni dif ferenziali. The second author
 gratefully acknowledges support from the Australian Research Council and the Austrian Science
 Fund (FWF), the latter via project P27072–N25. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Geometry of Almost Einstein (2,3,5) Distributions Article published earlier |
| spellingShingle | The Geometry of Almost Einstein (2,3,5) Distributions Sagerschnig, K. Willse, T. |
| title | 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_short | The Geometry of Almost Einstein (2,3,5) Distributions |
| title_sort | geometry of almost einstein (2,3,5) distributions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148555 |
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