Given a homogeneous pseudo-Riemannian space (G/H,⟨,⟩),\documentclass[12pt]{minimal}
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\begin{document}$$(G/H,\langle \ , \ \rangle),$$\end{document} a geodesic γ:I→G/H\documentclass[12pt]{minimal}
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\begin{document}$$\gamma :I\rightarrow G/H$$\end{document} is said to be two-step homogeneous if it admits a parametrization t=ϕ(s)\documentclass[12pt]{minimal}
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\begin{document}$$t=\phi (s)$$\end{document} (s affine parameter) and vectors X, Y in the Lie algebra g\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak{g}}$$\end{document}, such that γ(t)=exp(tX)exp(tY)·o\documentclass[12pt]{minimal}
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\begin{document}$$\gamma (t)=\exp (tX)\exp (tY)\cdot o$$\end{document}, for all t∈ϕ(I)\documentclass[12pt]{minimal}
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\begin{document}$$t\in \phi (I)$$\end{document}. 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 ⟨,⟩\documentclass[12pt]{minimal}
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\begin{document}$$\langle \ ,\ \rangle$$\end{document} on the unimodular Lie group SL(2,R)\documentclass[12pt]{minimal}
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\begin{document}$$SL(2,{{\mathbb{R}}})$$\end{document} such that (SL(2,R),⟨,⟩)\documentclass[12pt]{minimal}
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\begin{document}$$\big (SL(2,{{\mathbb{R}}}),\langle \ ,\ \rangle \big )$$\end{document} is a two-step g.o. space.
