Classification of curves on de Sitter plane

Abstract

In 1917, de Sitter used the modified Einstein equation and proposed a model of the Universe without physical matter, but with a cosmological constant. De Sitter geometry, as well as Minkowski geometry, is maximally symmetrical. However, de Sitter geometry is better suited to describe gravitational fields. It is believed that the real Universe was described by the de Sitter model in the very early stages of expansion (inflationary model of the Universe). This article is devoted to the problem of classification of regular curves on the de Sitter space. As a model of the de Sitter plane, the upper half-plane on which the metric is given is chosen. For this purpose, an algebra of differential invariants of curves with respect to the motions of the de Sitter plane is constructed. As it turned out, this algebra is generated by one second-order differential invariant (we call it by de Sitter curvature) and two invariant differentiations. Thus, when passing to the next jets, the dimension of the algebra of differential invariants increases by one. The concept of regular curves is introduced. Namely, a curve is called regular if the restriction of de Sitter curvature to it can be considered as parameterization of the curve. A theorem on the equivalence of regular curves with respect to the motions of the de Sitter plane is proved. The singular orbits of the group of proper motions are described.