Automorphisms of local order structures on hyperbolic manifolds (Q1109337)

A (partial) order is determined by the family of sets \(P_ x=\{y\geq x\}\). This leads in an obvious way to the notion of a local order in a topological space. A hyperbolic manifold M has by definition (the Poincaré model of) the hyperbolic space \(H_ n\) as simply connected covering space. The author considers a local order \(\Pi\) on M induced by an order P of \(H_ n\) which is invariant under a simply transitive motion group. \(\Pi\) is called elliptic if P is generated by a circular conus in the Poincaré model. Theorem. If \(\Pi\) is elliptic, U is open in M, and if f: \(U\to M\) is a \(\Pi\)-monomorphism, then f is locally conformal (with respect to the Minkowski metric).