On the behavior of geodesics of left-invariant sub-Riemannian metrics on the group $\operatorname{Aff}_{0}(\mathbb{R}) \times \operatorname{Aff}_{0}(\mathbb{R})$

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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.

Keywords

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

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