Closed geodesics in the tangent sphere bundle of a hyperbolic three-manifold

Let M be an oriented three-dimensional manifold of constant sectional curvature -1 and with positive injectivity radius, and T-1 M its tangent sphere bundle endowed with the canonical (Sasaki) metric. We describe explicitly the periodic geodesics of (TM)-M-1 in terms of the periodic geodesics of M: For a generic periodic geodesic (h. v) in (TM)-M-1, h is a periodic helix in M, whose axis is a periodic geodesic in M: the closing condition on (h, v) is given in terms of the horospherical radius of h and the complex length (length and holonomy) of its axis. As a corollary, we obtain that if two compact oriented hyperbolic three-manifolds have the same complex length spectrum (lengths and holonomies of periodic geodesics, with multiplicities). then their tangent sphere bundles are length isospectral, even if the manifolds are not isometric.