The Lorentzian oscillator group as a geodesic orbit space

On null-geodesic completions of Lorentzian manifolds

We study the question of local and global uniqueness of completions, based on null geodesics, of Lorentzian manifolds. We show local uniqueness of such boundary extensions. We give a necessary and sufficient condition for existence of unique maximal completions. The condition is verified in several situations of interest. This leads to existence and uniqueness of maximal spacelike conformal bou...

Some geometrical properties of the oscillator group

‎We consider the oscillator group equipped with‎ ‎a biinvariant Lorentzian metric‎. ‎Some geometrical properties of this space and the harmonicity properties of left-invariant vector fields on this space are determined‎. ‎In some cases‎, ‎all these vector fields are critical points for the energy functional‎ ‎restricted to vector fields‎. ‎Left-invariant vector fields defining harmonic maps are...

The Orbit Space of a Kleinian Group: Riley's Modest Example

In the previous paper, Robert Riley [4] and his computer file Poincaré found a fundamental domain for the action of a discrete group G of isometries of hyperbolic space H3 generated by three parabolics. In this paper, we show that the orbit space H3/G is homeomorphic to a complement 53 — k*, where k* is k union a point and where k is the (3,3,3) pretzel knot. Furthermore, HI 3/G is equipped wit...

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

A Null Geodesic Orbit Space Whose Null Orbits Require a Reparametrization

We exhibit an example of a homogeneous Lorentzian manifold G/H whose homogeneous geodesics form just the light-cone and a hyperplane in the tangent space. For all light-like homogeneous geodesics, the natural parameter of the orbit is not the affine parameter of the geodesic.