Tohoku Mathematical Journal
2001
March
SECOND SERIES VOL. 53, NO. 1

Tohoku Math. J.
53 (2001), 149-161

Title CLOSED GEODESICS IN THE TANGENT SPHERE BUNDLE OF A HYPERBOLIC THREE-MANIFOLD

Author Máximo Carreras and Marcos Salvai

(Received June 7, 1999, revised February 1, 2000)
Abstract. Let $M$ be an oriented three-dimensional manifold of constant sectional curvature $-1$ and with positive injectivity radius, and $T^1M$ its tangent sphere bundle endowed with the canonical (Sasaki) metric. We describe explicitly the periodic geodesics of $T^1M$ in terms of the periodic geodesics of $M$: For a generic periodic geodesic $(h,v)$ in $T^1M$, $h$ 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.

2000 Mathematics Subject Classification. Primary 53C22.

To the top of this page

Back to the Contents