Применение пакетов символьных вычислений к исследованию оператора одномерной кривизны на нередуктивных однородных псевдоримановых многообразиях

Abstract

The study of curvature operator properties, in particular, the one-dimensional curvature operator, is interesting for the understanding of the geometrical and topological structure of a homogeneous (pseudo)Riemannian manifold. In general case, this problem is quite difficult. Therefore, it is necessary to impose restrictions either on the class of manifolds or their dimension. An application of analytical calculation systems is possible if the dimension is finite. In this paper, mathematical and computer models for determining the components of the one-dimensional curvature operator and its spectrum (the set of eigenvalues) of non-reductive homogeneous (pseudo)Riemannian manifolds of a finite dimension are developed. The investigation of one-dimensional curvature operator spectrum on non-reductive homogeneous Lorentzian manifolds of dimension 4 is performed by implementing this algorithm in Maple software. Also, a symmetric operator with a matrix corresponding to a matrix of the one-dimensional curvature tensor is defined, and the problem of this operator possible signature existence on four-dimensional non-reductive homogeneous Lorentzian manifolds is studied.