Tohoku Mathematical Journal

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

Máximo Carreras and Marcos Salvai

Full-text: Open access

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

Permanent link to this document
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
Primary: 53C22: Geodesics [See also 58E10]
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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References

  • [1] A. BEARDON, Geometry of discrete groups, Grad. Texts in Math., Springer-Verlag, New York, 1983.
  • [2] H. GLUCK, Geodesies in the unit tangent bundle of a round sphere, Enseign. Math. 34 (1988), 233-246
  • [3] C. GORDON, Naturally reductive homogeneous Riemannian manifolds, Canad. J. Math. 37 (1985), 467-487
  • [4] C. GORDON AND Y. MAO, Comparisons of Laplace spectra, length spectra and geodesic flows of som Riemannian manifolds, Math. Res. Lett. 1 (1994), 677-688.
  • [5] T. KONNO AND S. TANNO, Geodesies and Killing vector fields on the tangent sphere bundle, Nagoya Math J. 151 (1998), 91-97.
  • [6] G. R. MEYERHOFF, The ortho-length spectrum for hyperbolic 3-manifolds, Quart. J. Math. Oxford Ser.(2 47 (1996), 349-359.
  • [7] A. REID, Isospectrality and commensurabilityof arithmetichyperbolic 2- and 3-manifolds, Duke Math. J. 6 (1992), 215-228.
  • [8] M. SALVAI, Spectra of unit tangent bundles of hyperbolic Riemann surfaces, Ann. Global Anal. Geom. 1 (1998), 357-370.
  • [9] M. SALVAI, On the Laplace and complex length spectra of compact locally symmetric spaces of negativ curvature, preprint.
  • [10] S. SASAKI, On the differential geometry of tangent bundles of Riemannian manifolds, Thoku Math. J. 1 (1958), 338-354.
  • [11] M. -F. ViGNERAS, Varietes riemanniennes isospectrales et non isometriques, Ann. of Math. (2) 112 (1980), 21-32.