Tohoku Mathematical Journal

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

Máximo Carreras and Marcos Salvai

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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

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

First available in Project Euclid: 3 May 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C22: Geodesics [See also 58E10]
Secondary: 58J53: Isospectrality


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.

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