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 ∞ Riemannian manifold of negative sectional curvature. Assume the stable or unstable foliation of the geodesic flow φ t : V → V on the unit tangent bundle V of M is C ∞ . Assume, moreover, that either (a) the sectional curvature of M satisfies −4 < K ≤ −1 or (b) the dimension of M is odd. Then the geodesic flow of M is C ∞ -isomorphic (i.e., conjugate under a C ∞ 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 φ 1 , is C ∞ -isomorphic to the flow , on the quotient Γ\ , where Γ is a subgroup of a real Lie group ⊂ Diffeo ( ) with Lie algebra is the geodesic flow on the unit tangent bundle of the complex hyperbolic space ℂH m , m = ½ dim M.