Contact and almost contact structures on the real extension of the Lobachevsky plane
In this article, we propose a group model G of a real extension of the Lobachevsky plane H2 × R . The group G is a Lie group of special-form matrices and a subgroup of the general linear group GL(3, R). It is proved that, on the group model of the real extension of the Lobachevsky plane, there is a...
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| Format: | Article |
| Language: | English |
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Kazan Federal University
2021-12-01
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| Series: | Учёные записки Казанского университета: Серия Физико-математические науки |
| Subjects: | |
| Online Access: | https://kpfu.ru/uz-eng-phm-2021-3-4-5.html |
| _version_ | 1826906384540631040 |
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| author | V.I. Pan’zhenskii A.O. Rastrepina |
| author_facet | V.I. Pan’zhenskii A.O. Rastrepina |
| author_sort | V.I. Pan’zhenskii |
| collection | DOAJ |
| description | In this article, we propose a group model G of a real extension of the Lobachevsky plane H2 × R . The group G is a Lie group of special-form matrices and a subgroup of the general linear group GL(3, R). It is proved that, on the group model of the real extension of the Lobachevsky plane, there is a unique left-invariant almost contact metric structure with the Riemannian metric of the direct product that is invariant with respect to the isometry group. The concept of a linear connection compatible with the distribution is introduced. All left-invariant linear connections for which the tensors of the almost contact metric structure (η, ξ, ϕ, g) are covariantly constant are found. Among the left-invariant differential 1-forms, a canonical form defining a contact structure on G is distinguished. The left-invariant contact metric connections are found. There is a unique left-invariant connection for which all tensors of the almost contact metric structure and the canonical contact form are covariantly constant. It is proved that this connection is compatible with the contact distribution in the sense that a single geodesic tangent to the contact distribution passes through each point in each contact direction. Parametric equations of geodesics of the given connection are found. It is also established that the Levi-Civita connection of the Riemannian metric of the direct product is not a connection compatible with the contact distribution. |
| first_indexed | 2024-04-11T02:02:35Z |
| format | Article |
| id | doaj.art-18789fdf560949e6b06f2be1f34b0185 |
| institution | Directory Open Access Journal |
| issn | 2541-7746 2500-2198 |
| language | English |
| last_indexed | 2025-02-17T08:47:23Z |
| publishDate | 2021-12-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета: Серия Физико-математические науки |
| spelling | doaj.art-18789fdf560949e6b06f2be1f34b01852025-01-02T20:43:09ZengKazan Federal UniversityУчёные записки Казанского университета: Серия Физико-математические науки2541-77462500-21982021-12-011633-429130310.26907/2541-7746.2021.3-4.291-303Contact and almost contact structures on the real extension of the Lobachevsky planeV.I. Pan’zhenskii0A.O. Rastrepina1Penza State University, Penza, 440026 RussiaPenza State University, Penza, 440026 RussiaIn this article, we propose a group model G of a real extension of the Lobachevsky plane H2 × R . The group G is a Lie group of special-form matrices and a subgroup of the general linear group GL(3, R). It is proved that, on the group model of the real extension of the Lobachevsky plane, there is a unique left-invariant almost contact metric structure with the Riemannian metric of the direct product that is invariant with respect to the isometry group. The concept of a linear connection compatible with the distribution is introduced. All left-invariant linear connections for which the tensors of the almost contact metric structure (η, ξ, ϕ, g) are covariantly constant are found. Among the left-invariant differential 1-forms, a canonical form defining a contact structure on G is distinguished. The left-invariant contact metric connections are found. There is a unique left-invariant connection for which all tensors of the almost contact metric structure and the canonical contact form are covariantly constant. It is proved that this connection is compatible with the contact distribution in the sense that a single geodesic tangent to the contact distribution passes through each point in each contact direction. Parametric equations of geodesics of the given connection are found. It is also established that the Levi-Civita connection of the Riemannian metric of the direct product is not a connection compatible with the contact distribution.https://kpfu.ru/uz-eng-phm-2021-3-4-5.htmllie groupcontact structurealmost contact structureleft-invariant connectioncontact geodesics |
| spellingShingle | V.I. Pan’zhenskii A.O. Rastrepina Contact and almost contact structures on the real extension of the Lobachevsky plane Учёные записки Казанского университета: Серия Физико-математические науки lie group contact structure almost contact structure left-invariant connection contact geodesics |
| title | Contact and almost contact structures on the real extension of the Lobachevsky plane |
| title_full | Contact and almost contact structures on the real extension of the Lobachevsky plane |
| title_fullStr | Contact and almost contact structures on the real extension of the Lobachevsky plane |
| title_full_unstemmed | Contact and almost contact structures on the real extension of the Lobachevsky plane |
| title_short | Contact and almost contact structures on the real extension of the Lobachevsky plane |
| title_sort | contact and almost contact structures on the real extension of the lobachevsky plane |
| topic | lie group contact structure almost contact structure left-invariant connection contact geodesics |
| url | https://kpfu.ru/uz-eng-phm-2021-3-4-5.html |
| work_keys_str_mv | AT vipanzhenskii contactandalmostcontactstructuresontherealextensionofthelobachevskyplane AT aorastrepina contactandalmostcontactstructuresontherealextensionofthelobachevskyplane |