Geodesic
ON THE GEOMETRY OF THE SPACE OF ORIENTED LINES OF THE HYPERBOLIC SPACE
Glasgow mathematical journal, Tome 49 (2007) no. 2, pp. 357-366

Voir la notice de l'article provenant de la source Cambridge University Press

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 .
DOI : 10.1017/S0017089507003710
Mots-clés : 53A55, 53C22, 53C35, 53C50, 53D25 
SALVAI, MARCOS. ON THE GEOMETRY OF THE SPACE OF ORIENTED LINES OF THE HYPERBOLIC SPACE. Glasgow mathematical journal, Tome 49 (2007) no. 2, pp. 357-366. doi: 10.1017/S0017089507003710
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[1] 1. Besse, A., Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihre Grenzgebiete no. 93 (Springer-Verlag, 1978). Google Scholar | DOI

[2] 2. Di Scala, A. and Olmos, C., The geometry of homogeneous submanifolds of hyperbolic space, Math. Z. 237 (2001), 199–209. Google Scholar | DOI

[3] 3. Guilfoyle, B. and Klingenberg, W., On the space of oriented affine lines in R3 , Archiv Math: (Basel). 82 (2004), 81–84. Google Scholar | DOI

[4] 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. Google Scholar | DOI

[5] 5. Hitchin, N. J., Monopoles and geodesics, Comm. Math. Phys. 83 (1982), 579–602. Google Scholar | DOI

[6] 6. Kanai, M., Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations, Ergodic Theory Dyn. Syst. 8 (1988), 215–239. Google Scholar | DOI

[7] 7. Keilhauer, G., A note on the space of geodesics, Rev. Unión Mat. Argent. 36 (1990), 164–173. Google Scholar

[8] 8. Palais, R., A global formulation of the Lie theory of transformation groups, Memoirs American Math. Soc. No. 22 (1957) Providence Google Scholar | DOI

[9] 9. Salvai, M., On the geometry of the space of oriented lines in Euclidean space, Manuscr. Math. 118 (2005), 181–189. Google Scholar | DOI

[10] 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] 11. Shepherd, M., Line congruences as surfaces in the space of lines. Diff. Geom. Appl. 10 (1999), 1–26. Google Scholar | DOI

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