Local Type I Metrics with Holonomy in G₂*
By [arXiv:1604.00528], a list of possible holonomy algebras for pseudo-Riemannian manifolds with an indecomposable torsion-free G₂*-structure is known. Here, indecomposability means that the standard representation of the algebra on R⁴,³ does not leave invariant any proper non-degenerate subspace. T...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2018 |
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Інститут математики НАН України
2018
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/209769 |
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| Cite this: | Local Type I Metrics with Holonomy in G₂* / A. Fino, I. Kath // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 14 назв. — англ. |
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Fino, A. Kath, I. 2025-11-26T11:32:07Z 2018 Local Type I Metrics with Holonomy in G₂* / A. Fino, I. Kath // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 14 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53C29; 53C50; 53C10 arXiv: 1705.00023 https://nasplib.isofts.kiev.ua/handle/123456789/209769 https://doi.org/10.3842/SIGMA.2018.081 By [arXiv:1604.00528], a list of possible holonomy algebras for pseudo-Riemannian manifolds with an indecomposable torsion-free G₂*-structure is known. Here, indecomposability means that the standard representation of the algebra on R⁴,³ does not leave invariant any proper non-degenerate subspace. The dimension of the socle of this representation is called the type of the Lie algebra. It is equal to one, two, or three. In the present paper, we use Cartan's theory of exterior differential systems to show that all Lie algebras of Type I from the list in [arXiv:1604.00528] can indeed be realised as the holonomy of a local metric. All these Lie algebras are contained in the maximal parabolic subalgebra p₁ that stabilises one isotropic line of R⁴,³. In particular, we realise p₁ by a local metric. We thank the referees for their valuable and constructive comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Local Type I Metrics with Holonomy in G₂* Article published earlier |
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| title |
Local Type I Metrics with Holonomy in G₂* |
| spellingShingle |
Local Type I Metrics with Holonomy in G₂* Fino, A. Kath, I. |
| title_short |
Local Type I Metrics with Holonomy in G₂* |
| title_full |
Local Type I Metrics with Holonomy in G₂* |
| title_fullStr |
Local Type I Metrics with Holonomy in G₂* |
| title_full_unstemmed |
Local Type I Metrics with Holonomy in G₂* |
| title_sort |
local type i metrics with holonomy in g₂* |
| author |
Fino, A. Kath, I. |
| author_facet |
Fino, A. Kath, I. |
| publishDate |
2018 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України |
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Article |
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By [arXiv:1604.00528], a list of possible holonomy algebras for pseudo-Riemannian manifolds with an indecomposable torsion-free G₂*-structure is known. Here, indecomposability means that the standard representation of the algebra on R⁴,³ does not leave invariant any proper non-degenerate subspace. The dimension of the socle of this representation is called the type of the Lie algebra. It is equal to one, two, or three. In the present paper, we use Cartan's theory of exterior differential systems to show that all Lie algebras of Type I from the list in [arXiv:1604.00528] can indeed be realised as the holonomy of a local metric. All these Lie algebras are contained in the maximal parabolic subalgebra p₁ that stabilises one isotropic line of R⁴,³. In particular, we realise p₁ by a local metric.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/209769 |
| citation_txt |
Local Type I Metrics with Holonomy in G₂* / A. Fino, I. Kath // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 14 назв. — англ. |
| work_keys_str_mv |
AT finoa localtypeimetricswithholonomying2 AT kathi localtypeimetricswithholonomying2 |
| first_indexed |
2025-12-07T19:48:42Z |
| last_indexed |
2025-12-07T19:48:42Z |
| _version_ |
1850886169239224320 |