On the Geodesic Orbit Property for Lorentz Manifolds

Abstract

The geodesic orbit property has been studied intensively for Riemannian manifolds. Geodesic orbit spaces are homogeneous and allow simplifications of many structural questions using the Lie algebra of the isometry group. Weakly symmetric Riemannian manifolds are geodesic orbit spaces. Here we define the property “naturally reductive” for pseudo-Riemannian manifolds and note that those manifolds are geodesic orbit spaces. A few years ago two of the authors proved that weakly symmetric pseudo-Riemannian manifolds are geodesic orbit spaces. In particular that result applies to pseudo-Riemannian Lorentz manifolds. Our main results are Theorems 7 and 8. In the Riemannian case the nilpotent isometry group for a geodesic orbit nilmanifold is abelian or 2-step nilpotent. Here we concentrate on the geodesic orbit property for Lorentz nilmanifolds G/H with $G = N \rtimes H$ and N nilpotent. When the metric is nondegenerate on $[{{\mathfrak {n}}},{{\mathfrak {n}}}]$, Theorem 7 shows that N either is at most 2-step nilpotent as in the Riemannian situation, or is 4-step nilpotent, but cannot be 3-step nilpotent. Surprisingly, Theorem 8 shows that N is at most 2-step nilpotent when the metric is degenerate on $[{{\mathfrak {n}}},{{\mathfrak {n}}}]$. Both theorems give additional structural information and specialize to naturally reductive and to weakly symmetric Lorentz nilmanifolds.