Geodesic flows of Riemannian or Finsler manifolds have been the only known contact Anosov flows. We show that even in dimension 3 the world of contact Anosov flows is vastly larger via a surgery construction near an E-transverse Legendrian link that encompasses both the Handel-Thurston and Goodman surgeries and that produces flows not topologically orbit equivalent to any algebraic flow. This includes examples on many hyperbolic 3-manifolds, any of which have remarkable dynamical and geometric properties. To the latter end we include a proof of a folklore theorem from 3-manifold topology: In the unit tangent bundle of a hyperbolic surface, the complement of a knot that projects to a filling geodesic is a hyperbolic 3-manifold.
We describe which Anosov flows on compact manifolds have
C
∞
{C^\infty }
stable and unstable distributions and a contact canonical
1
1
-form: up to finite coverings and up to a
C
∞
{C^\infty }
change of parameters, each of them is isomorphic to the geodesic flow on (the unit tangent bundle of) a compact locally symmetric space of strictly negative curvature.
We show how geodesics, Jacobi vector fields and flag curvature of a Finsler metric behave under Zermelo deformation with respect to a Killing vector field. We also show that Zermelo deformation with respect to a Killing vector field of a locally symmetric Finsler metric is also locally symmetric.
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