Cycles in de Sitter’s world

Abstract

A Lie cycle is a point in spherical geometry or an oriented spherical hypersphere. This definition as well as the definition of oriented contact between pairs of cycles are applied to de Sitter’s world. We determine the transformation group of the geometry of de Sitter cycles which consists of all permutations of the set ${\cal C}^{n}$ of de Sitter cycles which preserve oriented contact in both directions. We show that an arbitrary mapping $f:{\cal C}^{n}\ \rightarrow \ {\cal C}^{n}$ which satisfies c, c′ are in oriented contact ⇔ f(c), f(c′) are in oriented contact for all $c,c^{\prime}\in{\cal C}^{n}$ is already an element of the transformation group.