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ON THE GEOMETRY OF THE SPACE OF ORIENTED LINES OF THE HYPERBOLIC SPACE

Published online by Cambridge University Press:  09 August 2007

MARCOS SALVAI
MARCOS SALVAI*
Affiliation:
FAMAF – CIEM, Ciudad Universitaria, 5000 Córdoba, Argentina e-mail: salvai@mate.uncor.edu
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Abstract

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Let H be the n-dimensional hyperbolic space of constant sectional curvature –1 and let G be the identity component of the isometry group of H. We find all the G-invariant pseudo-Riemannian metrics on the space of oriented geodesics of H (modulo orientation preserving reparametrizations). We characterize the null, time- and space-like curves, providing a relationship between the geometries of and H. Moreover, we show that is Kähler and find an orthogonal almost complex structure on .

Keywords

53A5553C2253C3553C5053D25
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 49 , Issue 2 , May 2007 , pp. 357 - 366
Copyright
Copyright © Glasgow Mathematical Journal Trust 2007

References

REFERENCES

1. Besse, A., Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihre Grenzgebiete no. 93 (Springer-Verlag, 1978).CrossRefGoogle Scholar
2. Di Scala, A. and Olmos, C., The geometry of homogeneous submanifolds of hyperbolic space, Math. Z. 237 (2001), 199–209.CrossRefGoogle Scholar
3. Guilfoyle, B. and Klingenberg, W., On the space of oriented affine lines in R3 , Archiv Math: (Basel). 82 (2004), 81–84.CrossRefGoogle Scholar
4. Guilfoyle, B. and Klingenberg, W., An indefinite Kähler metric on the space of oriented lines, J. London Math. Soc. (2). 72 (2005), 497–509.CrossRefGoogle Scholar
5. Hitchin, N. J., Monopoles and geodesics, Comm. Math. Phys. 83 (1982), 579–602.CrossRefGoogle Scholar
6. Kanai, M., Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations, Ergodic Theory Dyn. Syst. 8 (1988), 215–239.CrossRefGoogle Scholar
7. Keilhauer, G., A note on the space of geodesics, Rev. Unión Mat. Argent. 36 (1990), 164–173.Google Scholar
8. Palais, R., A global formulation of the Lie theory of transformation groups, Memoirs American Math. Soc. No. 22 (1957) ProvidenceCrossRefGoogle Scholar
9. Salvai, M., On the geometry of the space of oriented lines in Euclidean space, Manuscr. Math. 118 (2005), 181–189.CrossRefGoogle Scholar
10. Salvai, M., Geodesics in the space of oriented lines in Euclidean space, Proceedings of Egeo 2005, Rev. Unión Mat. Argent. (2) 47 (2006), 109–114.Google Scholar
11. Shepherd, M., Line congruences as surfaces in the space of lines. Diff. Geom. Appl. 10 (1999), 1–26.CrossRefGoogle Scholar