Anosov flows with smooth foliations and rigidity of geodesic flows on three-dimensional manifolds of negative curvature

We consider Anosov flows on a 5-dimensional smooth manifold V that possesses an invariant symplectic form (transverse to the flow) and a smooth invariant probability measure λ. Our main technical result is the following: If the Anosov foliations are C ∞, then either (1) the manifold is a transversely locally symmetric space, i.e. there is a flow-invariant C ∞ affine connection ∇ on V such that ∇ R ≡ 0, where R is the curvature tensor of ∇, and the torsion tensor T only has nonzero component along the flow direction, or (2) its Oseledec decomposition extends to a C∞ splitting of TV (defined everywhere on V ) and for any invariant ergodic measure μ, there exists χ μ > 0 such that the Lyapunov exponents are −2χ μ , −χ μ , 0, χ μ , and 2χ μ , μ-almost everywhere. As an application, we prove: Given a closed three-dimensional manifold of negative curvature, assume the horospheric foliations of its geodesic flow are C ∞. Then, this flow is C ∞ conjugate to the geodesic flow on a manifold of constant negative curvature.