On the behavior of geodesics of left-invariant sub-Riemannian metrics on the group $ \operatorname{Aff}_{0}(\mathbb{R}) \times \operatorname{Aff}_{0}(\mathbb{R}) $
In this paper, we study geodesics of left-invariant sub-Riemannian metrics on the Cartesian square of a connected two-dimensional non-commutative Lie group, where the metric is determined by the inner product on a two-dimensional generating subspace of the corresponding Lie algebra. It is proven tha...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2025-01-01
|
| Series: | Electronic Research Archive |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2025010 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850216109457801216 |
|---|---|
| author | Yuriĭ G. Nikonorov Irina A. Zubareva |
| author_facet | Yuriĭ G. Nikonorov Irina A. Zubareva |
| author_sort | Yuriĭ G. Nikonorov |
| collection | DOAJ |
| description | In this paper, we study geodesics of left-invariant sub-Riemannian metrics on the Cartesian square of a connected two-dimensional non-commutative Lie group, where the metric is determined by the inner product on a two-dimensional generating subspace of the corresponding Lie algebra. It is proven that the system of equations for geodesics of such a sub-Riemannian metric is not completely integrable in the class of meromorphic functions. Important qualitative characteristics of the corresponding geodesics are found, thus proving the complexity of their behavior in general. |
| format | Article |
| id | doaj-art-e5e5f70d4a2640a291278214cc63a0e2 |
| institution | OA Journals |
| issn | 2688-1594 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj-art-e5e5f70d4a2640a291278214cc63a0e22025-08-20T02:08:25ZengAIMS PressElectronic Research Archive2688-15942025-01-0133118120910.3934/era.2025010On the behavior of geodesics of left-invariant sub-Riemannian metrics on the group $ \operatorname{Aff}_{0}(\mathbb{R}) \times \operatorname{Aff}_{0}(\mathbb{R}) $Yuriĭ G. Nikonorov0Irina A. Zubareva1Southern Mathematical Institute of VSC RAS, Vladikavkaz 362025, RussiaOmsk Department of Sobolev Institute of Mathematics of SB RAS, Omsk 644099, RussiaIn this paper, we study geodesics of left-invariant sub-Riemannian metrics on the Cartesian square of a connected two-dimensional non-commutative Lie group, where the metric is determined by the inner product on a two-dimensional generating subspace of the corresponding Lie algebra. It is proven that the system of equations for geodesics of such a sub-Riemannian metric is not completely integrable in the class of meromorphic functions. Important qualitative characteristics of the corresponding geodesics are found, thus proving the complexity of their behavior in general.https://www.aimspress.com/article/doi/10.3934/era.2025010extremalgenerating subspace of a lie algebrahamiltonian systemkovalevskaya exponentsleft-invariant sub-riemannian metricnon-commutative two-dimensional lie groupsub-riemannian geodesic |
| spellingShingle | Yuriĭ G. Nikonorov Irina A. Zubareva On the behavior of geodesics of left-invariant sub-Riemannian metrics on the group $ \operatorname{Aff}_{0}(\mathbb{R}) \times \operatorname{Aff}_{0}(\mathbb{R}) $ Electronic Research Archive extremal generating subspace of a lie algebra hamiltonian system kovalevskaya exponents left-invariant sub-riemannian metric non-commutative two-dimensional lie group sub-riemannian geodesic |
| title | On the behavior of geodesics of left-invariant sub-Riemannian metrics on the group $ \operatorname{Aff}_{0}(\mathbb{R}) \times \operatorname{Aff}_{0}(\mathbb{R}) $ |
| title_full | On the behavior of geodesics of left-invariant sub-Riemannian metrics on the group $ \operatorname{Aff}_{0}(\mathbb{R}) \times \operatorname{Aff}_{0}(\mathbb{R}) $ |
| title_fullStr | On the behavior of geodesics of left-invariant sub-Riemannian metrics on the group $ \operatorname{Aff}_{0}(\mathbb{R}) \times \operatorname{Aff}_{0}(\mathbb{R}) $ |
| title_full_unstemmed | On the behavior of geodesics of left-invariant sub-Riemannian metrics on the group $ \operatorname{Aff}_{0}(\mathbb{R}) \times \operatorname{Aff}_{0}(\mathbb{R}) $ |
| title_short | On the behavior of geodesics of left-invariant sub-Riemannian metrics on the group $ \operatorname{Aff}_{0}(\mathbb{R}) \times \operatorname{Aff}_{0}(\mathbb{R}) $ |
| title_sort | on the behavior of geodesics of left invariant sub riemannian metrics on the group operatorname aff 0 mathbb r times operatorname aff 0 mathbb r |
| topic | extremal generating subspace of a lie algebra hamiltonian system kovalevskaya exponents left-invariant sub-riemannian metric non-commutative two-dimensional lie group sub-riemannian geodesic |
| url | https://www.aimspress.com/article/doi/10.3934/era.2025010 |
| work_keys_str_mv | AT yuriignikonorov onthebehaviorofgeodesicsofleftinvariantsubriemannianmetricsonthegroupoperatornameaff0mathbbrtimesoperatornameaff0mathbbr AT irinaazubareva onthebehaviorofgeodesicsofleftinvariantsubriemannianmetricsonthegroupoperatornameaff0mathbbrtimesoperatornameaff0mathbbr |