On the Ergodicity of Frame Flows

Let V be a closed (i.e. compact without boundary) connected n-dimensional Riemannian manifold of class C 3. Denote by Stk(V ) the space of the orthonormal tangent k-frames of V. There is a natural fibration Stk(V)--,V associated to the tangent bundle T(V); its fiber is the Stiefel manifold St~ of the orthonormal kframes in IR". According to our notations Stl(V ) is the bundle of the unit tangent vectors. The Riemannian structure on V induces an R-action in StI(V) called the geodesic flow. Consider the natural projection Stk(V)-~Stl(V ) (its fiber is StT,-~) and lift the geodesic flow to a flow in Stk(V ) as follows: Take a frame (el, e 2 . . . . , ek) at a point v~V, eIET~(V). The geodesic flow sends e I to vectors tangent to the geodesic determined by e 1. The lifted flow sends our original frame to the frames parallel to it along the geodesic. This flow in Stk(V ) is called the k-frame flow.