On the Geometric Orbit Property for Lorentz Manifolds

TL;DR

This paper investigates the structure of Lorentz nilmanifolds with the geodesic orbit property, revealing bounds on nilpotency steps and structural conditions, extending known Riemannian results to pseudo-Riemannian settings.

Contribution

It establishes new bounds on the nilpotency step of the nilpotent group in Lorentz nilmanifolds with the geodesic orbit property, especially distinguishing between degenerate and nondegenerate metric cases.

Findings

In nondegenerate cases, nilpotent groups are at most 2- or 4-step nilpotent.

In degenerate cases, nilpotent groups are at most 2-step nilpotent.

Results extend Riemannian geodesic orbit properties to Lorentz pseudo-Riemannian 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 "naturally reductive" for pseudo-Riemannian manifolds and note that they 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 these results apply to pseudo-Riemannian Lorentz manifolds. There our main results are Theorems 4.2 and 5.1. In the Riemannian case the nilpotent isometry group for a geodesic orbit nilmanifold is abelian or $2$-step nilpotent. Examples show that this fails dramatically in the pseudo-Riemannian case. 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 4.2 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. Examples show that these bounds are the best possible. Surprisingly, Theorem 5.1 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. Key Words: Geodesic Orbit Space; Lorentz nilmanifold; Weakly Symmetric Space; Naturally Reductive Space; Pseudo-Riemannian Manifold.