On the geometry of the space of oriented lines of Euclidean space

SALVAI, MARCOS

MANUSCRIPTA MATHEMATICA

SPRINGER

Año: 2005 vol. 118 p. 181 - 189

0025-2611

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. We also give explicitly the geodesics and the geometric meaning of them being null, time- or space-like. Besides, we prove that the given metrics are Kahler or nearly Kahler if n = 3 or n = 7, respectively.