Closed geodesics in the tangent sphere bundle of a hyperbolic manifold

SALVAI, MARCOS; CARRERAS, MÁXIMO

TOHOKU MATHEMATICAL JOURNAL

TOHOKU UNIVERSITY

Año: 2001 vol. 53 p. 149 - 161

0040-8735

Let M be an oriented three-dimensional manifold of constant sec- tional curvature -1 and with positive injectivity radius, and T1M its tangent sphere bundle endowed with the canonical (Sasaki) met- ric. We describe explicitly the periodic geodesics of T1M in terms of the periodic geodesics of M: For a generic periodic geodesic (h,v) in T1M; 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 horospher- ical 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 tan- gent sphere bundles are length isospectral, even if the manifolds are not isometric.