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. 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, there exists > 0 such that the Lyapunov exponents are −2, −, 0,, and 2, -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.