Space of Geodesics in Hyperbolic Spaces and Lorentz Numbers

Abstract

In this note, we will study about the space of oriented geodesics in hyperbolic spaces H^n. It is well-known that the space of oriented geodesics (i.e., oriented great circles) in spheres S^n is identified with oriented real 2-plane Grassmannian G^2(R^<n+1>) and complex quadric Q^n. We will show that the space of oriented geodesics in H^n is also given similarly by using Lorentz numbers. Oriented real 2-plane Grassmannian plays important roles among differential geometry of submanifolds. For example, let f be an immersion from a Riemann surface ∑ to the Euclidean space R^<n+1>. Then the Gauss map γ from ∑ to the Grassmannian G^2(R^<n+1>) of oriented 2-plane in R^<n+1> of f is anti-holomorphic (resp. holomorphic) if and only if the immersion f is minimal (resp. totally umbilical). Here we will remark that similar results valid for timelike surfaces in Lorentz space R^<n+1> without proof.