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}) $

Yuriĭ G. Nikonorov; Irina A. Zubareva; Yuriĭ G. Nikonorov; Irina A. Zubareva

doi:10.3934/era.2025010

Electronic Research Archive

2025, Volume 33, Issue 1: 181-209. doi: 10.3934/era.2025010

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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}) $

Yuriĭ G. Nikonorov ^{1
,
,},
Irina A. Zubareva ²

1.
Southern Mathematical Institute of VSC RAS, Vladikavkaz 362025, Russia
2.
Omsk Department of Sobolev Institute of Mathematics of SB RAS, Omsk 644099, Russia

Received: 18 November 2024 Revised: 17 December 2024 Accepted: 02 January 2025 Published: 20 January 2025

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.
Citation: 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}) $[J]. Electronic Research Archive, 2025, 33(1): 181-209. doi: 10.3934/era.2025010

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Abstract

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.

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