Two-step homogeneous geodesics in pseudo-Riemannian manifolds

Given a homogeneous pseudo-Riemannian space (G/H,⟨,⟩), a geodesic γ: I→ G/ H is said to be two-step homogeneous if it admits a parametrization t= ϕ(s) (s affine parameter) and vectors X, Y in the Lie algebra g, such that γ(t) = exp (tX) exp (tY) · o, for all t∈ ϕ(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.