Two-step homogeneous geodesics in pseudo-Riemannian manifolds

Abstract

Given a homogeneous pseudo-Riemannian space $$(G/H,\\langle \\ , \\ \\rangle),$$\n a geodesic $$\\gamma :I\\rightarrow G/H$$\n is said to be two-step homogeneous if it admits a parametrization $$t=\\phi (s)$$\n (s affine parameter) and vectors X,\xa0Y in the Lie algebra $${\\mathfrak{g}}$$\n , such that $$\\gamma (t)=\\exp (tX)\\exp (tY)\\cdot o$$\n , for all $$t\\in \\phi (I)$$\n . 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 $$\\langle \\ ,\\ \\rangle$$\n on the unimodular Lie group $$SL(2,{{\\mathbb{R}}})$$\n such that $$\\big (SL(2,{{\\mathbb{R}}}),\\langle \\ ,\\ \\rangle \\big )$$\n is a two-step g.o. space.