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2001 Closed geodesics in the tangent sphere bundle of a hyperbolic three-manifold
Máximo Carreras, Marcos Salvai
Tohoku Math. J. (2) 53(1): 149-161 (2001). DOI: 10.2748/tmj/1178207537

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.

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Máximo Carreras. Marcos Salvai. "Closed geodesics in the tangent sphere bundle of a hyperbolic three-manifold." Tohoku Math. J. (2) 53 (1) 149 - 161, 2001. https://doi.org/10.2748/tmj/1178207537

Information

Published: 2001
First available in Project Euclid: 3 May 2007

zbMATH: 1020.53020
MathSciNet: MR2002E:53050
Digital Object Identifier: 10.2748/tmj/1178207537

Subjects:
Primary: 53C22
Secondary: 58J53

Rights: Copyright © 2001 Tohoku University

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