Groups of Motions of Properly Three-Dimensional Helmholtz Geometry and Simplical Three-Dimensional Geometry of the III Type
УДК 514.1
Abstract
The main tasks of the theory of phenomenologically symmetric (PS) geometries (geometries of local maximum mobility) are a complete classification of such geometries, derivation of the equation of phenomenological symmetry and finding groups of motions for the geometries. PS geometry is defined on a manifold by a function of a pair of points. Phenomenological symmetry of three-dimensional PS geometries lies in the presence of functional relation between the values of a pair of points for all pairs of five arbitrary points. Their classification was first built by V.Kh. Lev and later supplemented by V.A. Kyrov with simplicial type III geometry. Methods for establishing group symmetry of PS geometry contain a method of solving functional equations on a set of motions developed for two-dimensional and some three-dimensional PS geometries and the exponential mapping method.This paper describes the process of finding explicit expressions for the groups of motions with the methodof exponential mapping for the properly Helmholtz and simplicial type III three-dimensional PS geometries.These calculations are made using the apparatus of complex analysis and formulated as a separate theorem. These groups are actions of three Lie groups SL2(C)R on the space R3.
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Copyright (c) 2019 Рада Богданова
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