## Tohoku Mathematical Journal

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

#### 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.

#### Article information

Source
Tohoku Math. J. (2), Volume 53, Number 1 (2001), 149-161.

Dates
First available in Project Euclid: 3 May 2007

https://projecteuclid.org/euclid.tmj/1178207537

Digital Object Identifier
doi:10.2748/tmj/1178207537

Mathematical Reviews number (MathSciNet)
MR2002e:53050

Zentralblatt MATH identifier
1020.53020

Subjects
Secondary: 58J53: Isospectrality

#### Citation

Carreras, Máximo; Salvai, Marcos. Closed geodesics in the tangent sphere bundle of a hyperbolic three-manifold. Tohoku Math. J. (2) 53 (2001), no. 1, 149--161. doi:10.2748/tmj/1178207537. https://projecteuclid.org/euclid.tmj/1178207537

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