Geodesic
Left-invariant metrics of some three-dimensional Lie groups
Matematičeskie zametki SVFU, Tome 30 (2023), pp. 24-36.

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Mikhailichenko constructed a complete classification of two-dimensional geometries of maximum mobility, which contains, in addition to well-known geometries, also three geometries of the Helmholtz type (actually Helmholtz, pseudo-Helmholtz, and dual Helmholtz). Each of these geometries is specified by a function of a pair of points (an analogue of the Euclidean distance) and is a geometry of local maximum mobility, that is, it allows a three-parameter group of movements. The groups of motions of these geometries are uniquely associated with non-unimodular matrix three-dimensional Lie groups, the study of which is the subject of this article. In this work, left-invariant metrics of the studied matrix Lie groups are constructed, and Levi-Civita connections are found, as well as curvature on these Lie groups. Geodesics on such Lie groups are studied.
Keywords: geometry of local maximum mobility, left-invariant Riemannian metrics, curvature, geodesic. 
@article{SVFU_2023_30_a2,
     author = {V. A. Kyrov},
     title = {Left-invariant metrics of some three-dimensional {Lie} groups},
     journal = {Matemati\v{c}eskie zametki SVFU},
     pages = {24--36},
     publisher = {mathdoc},
     volume = {30},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVFU_2023_30_a2/}
}
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V. A. Kyrov. Left-invariant metrics of some three-dimensional Lie groups. Matematičeskie zametki SVFU, Tome 30 (2023), pp. 24-36. http://geodesic.mathdoc.fr/item/SVFU_2023_30_a2/