HOMOGENEOUS SPACES OF NONPOSITIVE CURVATURE AND THEIR GEODESIC-FLOW

Consider the geodesic flow on the unit tangent bundle SH of a 1-connec ted, irreducible homogeneous space H of nonpositive curvature. We prov e that any flow invariant, isometry invariant C-0-function on SH is ne cessarily constant, unless H is symmetric of higher rank. As the main applications, we obtain rigidity and partial classification results fo r spaces H whose geodesic symmetries are (asymptotically) volume-prese rving.