Features of the geometry of the five-dimensional pseudo-Euclidean space of index two

Abstract

The article is devoted to the study of the geometry of subspaces of a five-dimensional pseudo-Euclidean space. This space is attractive because all kinds of semi-Euclidean, semi-pseudo-Euclidean, hyperbolic three-dimensional spaces with projective metrics are realized in its subspaces. In the sphere of the imaginary radius of space, de Sitter space is realized. Here there is a space with projective metrics in the sense of Cayley-Klein. It is a three-dimensional space with a metric that preserves space on itself when mapped linearly. The corresponding linear transformation is called the motion of this space. An interpretation of de Sitter space in a four-dimensional pseudo-Euclidean space is proved. Studies have confirmed that in subspaces of space , in addition to elliptic spaces, there is a geometry of three-dimensional spaces with projective metrics. De Sitter space of the second kind is also realized in the sphere of imaginary radius. De Sitter space is a geodesic mapping in four-dimensional Minkowski space.