The Structure of Geodesic Orbit Lorentz Nilmanifolds

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.