Classification of Petrov Homogeneous Spaces
V.V. Obukhov

TL;DR
This paper completes the Petrov classification for certain homogeneous spaces, focusing on the structure of vector fields related to Killing vectors and their role in simplifying physical equations.
Contribution
It provides a complete classification of vector fields associated with Petrov type VIII homogeneous spaces where the group acts simply transitively.
Findings
Refined Petrov classification for type VIII homogeneous spaces.
Complete classification of vector fields for spaces with simply transitive G_3.
Simplification of mathematical physics equations in these spaces.
Abstract
In this paper the final stage of the Petrov classification is carried out. As it is known, the Killing vector fields specify infinitesimal transformations of the group of motions of space . In the case when in the homogeneous space the group of motions acts simply transitive, the geometry of the non-isotropic hypersurface is determined by the geometry of the transitivity space of the group . In this case, the metric tensor of the space can be given by a nonholonomic reper consisting of three independent vectors , which define the generators of the group of finite transformations in the space . The representation of the metric tensor of spaces by means of vector fields has a great physical meaning and allows to simplify substantially the equations of mathematical physics in such spaces. Therefore,…
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