Two-step homogeneous geodesics in pseudo-Riemannian manifolds

Abstract

Given a homogeneous pseudo-Riemannian space $$(G/H,\langle \ , \ \rangle),$$ a geodesic $$\gamma :I\rightarrow G/H$$ 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 $${\mathfrak{g}}$$ , such that $$\gamma (t)=\exp (tX)\exp (tY)\cdot o$$ , for all $$t\in \phi (I)$$ . 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 $$\langle \ ,\ \rangle$$ on the unimodular Lie group $$SL(2,{{\mathbb{R}}})$$ such that $$\big (SL(2,{{\mathbb{R}}}),\langle \ ,\ \rangle \big )$$ is a two-step g.o. space.