Left-invariant metrics of some three-dimensional Lie groups
https://doi.org/10.25587/2411-9326-2023-4-24-36
Abstract
Mikhailichenko constructed a complete classification of two-dimensional geometries of maximum mobility, which contains, in addition to well-known geometries, 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.
About the Author
V. A. Kyrov
Gorno-Altaisk State University
Russian Federation
Russian Federation
Vladimir A. Kyrov
Lenkina st., 1, Gorno-Altaisk 649000
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Review
For citations:
Kyrov V.A. Left-invariant metrics of some three-dimensional Lie groups. Mathematical notes of NEFU. 2023;30(4):24-36. (In Russ.) https://doi.org/10.25587/2411-9326-2023-4-24-36
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