Two-step homogeneous geodesics in pseudo-Riemannian manifolds

Given a homogeneous pseudo-Riemannian space (G/H, < , >), a geodesic gamma : I -> G/H is said to be two-step homogeneous if it admits a parametrization t = phi(s) (s affine parameter) and vectors X, Y in the Lie algebra g, such that gamma(t) = exp(tX) exp(tY) . o, for all t is an element of phi(I). As such, two-step homogeneous geodesics are a natural generalization of homogeneous geodesics (i.e., geodesics which are orbits of a one-parameter group of isometries). We obtain characterizations of two-step homogeneous geodesics, both for reductive homogeneous spaces and in the general case, and undertake the study of two-step g.o. spaces, that is, homogeneous pseudo-Riemannian manifolds all of whose geodesics are two-step homogeneous. We also completely determine the left-invariant metrics <, > on the unimodular Lie group SL(2, R) such that (SL(2, R), <, >) is a two-step g.o. space.