Pseudo-Riemannian Geodesic Orbit Nilmanifolds of Signature \(\varvec{(n-2,2)}\)

The geodesic orbit property is useful and interesting in itself, and it plays a key role 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 extend Riemannian and Lorentz results to trans-Lorentz nilmanifolds. Those are the geodesic orbit pseudo Riemannian manifolds \(M = G/H\) $M=G/H$ of signature \((n-2,2)\) $(n-2,2)$ such that a nilpotent analytic subgroup of G is transitive on M. For that we suppose that there is a reductive decomposition \({\mathfrak {g}}= {\mathfrak {h}}\oplus {\mathfrak {n}}\text { (vector space direct sum) with } [{\mathfrak {h}},{\mathfrak {n}}] \subset {\mathfrak {n}}\) $\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{n}\text{(vector space direct sum) with}[\mathfrak{h},\mathfrak{n}]\subset \mathfrak{n}$ and \({\mathfrak {n}}\) $\mathfrak{n}$ nilpotent. When the metric is nondegenerate on \([{\mathfrak {n}},{\mathfrak {n}}]\) $[\mathfrak{n},\mathfrak{n}]$ we show that \({\mathfrak {n}}\) $\mathfrak{n}$ is abelian or 2-step nilpotent. That is the same result as for geodesic orbit Riemannian and Lorentz nilmanifolds. When the metric is degenerate on \([{\mathfrak {n}},{\mathfrak {n}}]\) $[\mathfrak{n},\mathfrak{n}]$ we show that \({\mathfrak {n}}\) $\mathfrak{n}$ is a double extension of a geodesic orbit nilmanifold of either Riemannian or Lorentz signature.