New geometric examples of Anosov actions

We study a new class of Anosov actions (in the sense of Hirsch, Pugh and Shub) of reductive Lie groups, which are related to Riemannian symmetric spaces of non-compact type. The orbits of these actions can be identified with unions of parallel geodesics and the resulting orbit spaces are symplectic manifolds. For symmetric spaces of rank 1 all actions coincide with the geodesic flow.