Geodesics as deformations of one-parameter subgroups in homogeneous manifolds

Abstract

We solve explicitly the geodesic equation for a wide class of (pseudo)-Riemannian homogeneous manifolds (G/H,m), including those with G compact, as well as non-compact semisimple Lie groups, under a simple algebraic condition for the metric m. We prove that these manifolds are geodesically complete and their geodesics are orbits in G/H of a product of N one-parameter subgroups of G, where N is a natural number. This is intimately related to the fact that the metrics m can be regarded as N-parameter deformations of a Riemannian metric in G/H with special symmetries. We show that there exists a wealth of metrics having the aforementioned type of geodesics; those metrics are constructed from any Lie subgroup series of the form H < H_1 < ... < H_{N-1} < G.