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

Let M be an oriented three-dimensional manifold of constant sectional curva ture -1 and with positive injectivity radius, and T-1 M its tangent sphere bundle endowed with the canonical (Sasaki) metric. We describe explicitly t he 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 i n 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 c ompact oriented hyperbolic three-manifolds have the same complex length spe ctrum (lengths and holonomies of periodic geodesics, with multiplicities). then their tangent sphere bundles are length isospectral, even if the manif olds are not isometric.