On the geometry of hyperbolic cycles

A hyperbolic cycle is a point or an oriented hypersphere of an n-dimensional hyperbolic space. We determine all surjective mappings from the set of hyperbolic cycles to itself which preserve oriented contact between pairs of hyperbolic cycles in both directions (Theorem 1), and all bijections of the set of hyperbolic cycles such that images and pre-images of hyperbolic spears resp. oriented horo-hyperspheres resp. oriented equidistant hypersurfaces are hyperbolic spears resp. oriented horo-hyperspheres resp. oriented equidistant hypersurfaces (Theorems 2 and 3). Two cyclographic projections and connections to Lie geometry and Robertson–Walker spacetimes are studied.