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.