Homogeneous geodesics in pseudo-Riemannian nilmanifolds

Abstract

We study the geodesic orbit property for nilpotent Lie groups $N$ when endowed with a pseudo-Riemannian left-invariant metric. We consider this property with respect to different groups acting by isometries. When $N$ acts on itself by left-translations we show that it is a geodesic orbit space if and only if the metric is bi-invariant. Assuming $N$ is 2-step nilpotent and with non-degenerate center we give algebraic conditions on the Lie algebra $\mathfrak n$ of $N$ in order to verify that every geodesic is the orbit of a one-parameter subgroup of $N\rtimes\operatorname{Auto}(N)$. In addition we present an example of an almost g.o. space such that for null homogeneous geodesics, the natural parameter of the orbit is not always the affine parameter of the geodesic.