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The main result is that if an Anosov flow in a closed hyperbolic threemanifold is not R-covered, then the flow is a quasigeodesic flow. We also provethat if a hyperbolic three manifold supports an Anosov flow, then up to adouble cover it supports a quasigeodesic flow. We prove the continuousextension property for the stable and unstable foliations of any Anosov flow ina closed hyperbolic three manifold, and the existence of group invariant Peanocurves associated with any such flow.

Non R-covered Anosov flows in hyperbolic 3-manifolds are quasigeodesic