Left-invariant metrics of some three-dimensional Lie groups

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.