The Structure of Geodesic Orbit Lorentz Nilmanifolds

TL;DR

This paper analyzes the structure of geodesic orbit Lorentz nilmanifolds, revealing conditions under which the nilpotent Lie algebra is abelian, 2-step nilpotent, or a Lorentz double extension, expanding understanding of indefinite metric geometries.

Contribution

It provides a major structural classification of geodesic orbit Lorentz nilmanifolds, identifying conditions on the nilpotent Lie algebra and constructing examples with unbounded nilpotency steps.

Findings

When the metric is nondegenerate on [n,n], n is abelian or 2-step nilpotent.

When the metric is degenerate on [n,n], n is a Lorentz double extension.

Examples show the number of nilpotency steps can be unbounded.

Abstract

The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz manifolds. Here we carry out a major step in the structural analysis of geodesic orbit Lorentz nilmanifolds. Those are the geodesic orbit Lorentz manifolds $M = G/H$ such that a nilpotent analytic subgroup of $G$ is transitive on $M$. Suppose that there is a reductive decomposition $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{n}$ (vector space direct sum) with $\mathfrak{n}$ nilpotent. When the metric is nondegenerate on $[\mathfrak{n},\mathfrak{n}]$ we show that $\mathfrak{n}$ is abelian or 2-step nilpotent (this is the same result as for geodesic orbit Riemannian nilmanifolds), and when the metric is degenerate on $[\mathfrak{n},\mathfrak{n}]$ we show that $\mathfrak{n}$ is a Lorentz double extension corresponding to a geodesic orbit Riemannian nilmanifold. In the latter case we construct examples to show that the number of nilpotency steps is unbounded.