The Structure of Geodesic Orbit Lorentz Nilmanifolds

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