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 ( t X ) exp ( t Y ) · 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 S L ( 2 , R ) such that ( S L ( 2 , R ) , ⟨ , ⟩ ) is a two-step g.o. space.