On the geometry of the space of oriented lines of Euclidean space
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We prove that the space of all oriented lines of the n-dimensional Euclidean space admits a pseudo-Riemannian metric which is invariant by the induced transitive action of a connected closed subgroup of the group of Euclidean motions, exactly when n=3 or n=7 (as usual, we consider Riemannian metrics as a particular case of pseudo-Riemannian ones). Up to equivalence, there are two such metrics for each dimension, and they are of split type and complete. Besides, we prove that the given metrics are Kähler or nearly Kähler if n=3 or n=7, respectively.
Mathematics Subject Classification (2000)53B30 53B35 53C22 53C30 22F30 32M10 32Q15
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- 1.Borel, A.: Some remarks about Lie groups transitive on spheres and tori. Bull. Amer. Math. Soc. 55, 580–587 (1949)Google Scholar
- 3.Guilfoyle B., Klingenberg W.: An indefinite Kähler metric on the space of oriented lines. To appear in J. London Math. Soc.Google Scholar
- 4.Harvey, F.R.: Spinors and calibrations. Perspectives in Mathematics, vol. 9. Boston: Academic Press, Inc. 1990Google Scholar
- 6.Kobayashi S., Nomizu, K.: Foundations of differential geometry. Vol. II. New York-London-Sydney: Interscience Publishers, 1969Google Scholar
- 7.Montgomery, D.; Samelson, H.: Transformation groups of spheres. Ann. Math. 44, 454–470 (1943)Google Scholar
- 9.Salvai, M.: On the geometry of the space of geodesics of a symmetric space of negative curvature. PreprintGoogle Scholar