To the question of local extension of the parallel translations group of three-dimensional space

V. A. Kyrov

Vladikavkazskij matematičeskij žurnal, Tome 23 (2021) no. 1, pp. 32-42

Résumé

In modern geometry, the problem of extending a transitive Lie group $G$ acting in the manifold $M$ is topical. By an extension of a transitive Lie group $G$ we mean a Lie group $G_1$ containing $G$ as a Lie subgroup and also transitive on $M$, and the restriction of this transitive action to $G$ gives the original transitive action of the Lie group $G$. In particular, we can talk about the extension of the group of parallel translations of the three-dimensional space $R^3$. In this paper, we pose the problem of finding all locally doubly transitive extensions of the parallel translation group of a three-dimensional space. This problem is reduced to computing the Lie algebras of locally doubly transitive extensions of the parallel translation group. Basic operators of such Lie algebras are found from solutions of singular systems of three differential equations. It is proved that the matrices of the coefficients of these systems of differential equations commute with each other. The first matrix is reduced to Jordan form, and the other two matrices are forgiven using commutativity and applying admissible transformations. As a result, we have six types of Lie algebras. A separate work will be devoted to finding explicit forms of such Lie algebras and the corresponding local Lie groups of transformations of three-dimensional space.