GEODESIC-FLOWS ON MANIFOLDS OF NEGATIVE CURVATURE WITH SMOOTH HOROSPHERIC FOLIATIONS

We improve and extend a result due to M. Kanai about rigidity of geodesic flows on closed Riemannian manifolds of negative curvature whose stable or unstable horospheric foliation is smooth. More precisely, the main results proved here are: (1) Let M be a closed C(infinity) Riemannian manifold of negative sectional curvature. Assume the stable or unstable foliation of the geodesic flow phi(tau): V --> V on the unit tangent bundle V of M is C(infinity). Assume, moreover, that either (a) the sectional curvature of M satisfies -4 < K less-than-or-equal-to -1 or (b) the dimension of M is odd. Then the geodesic flow of M is C(infinity)-isomorphic (i.e., conjugate under a C(infinity) diffeomorphism between the unit tangent bundles) to the geodesic flow on a closed Riemannian manifold of constant negative curvature. (2) For M as above, assume instead of (a) or (b) that dim M = 2(mod 4). Then either the above conclusion holds or phi(t) is C(infinity)-isomorphic to the flow phi(t) on the quotient GAMMA\V, where GAMMA is a subgroup of a real Lie group GBAR subset-of Diffeo (V) with Lie algebra su(1, m) + R and phi(t): V --> V is the geodesic flow on the unit tangent bundle of the complex hyperbolic space CH(m), m = 1/2 dim M.