Geodesics of the space of oriented lines of the Euclidean space

SALVAI, MARCOS

REVISTA DE LA UNIóN MATEMáTICA ARGENTINA

UNION MATEMATICA ARGENTINA

Año: 2006 vol. 47 p. 109 - 114

0041-6932

For n = 3 or n = 7 let Tn be the space of oriented lines in Rn. In a previous article we characterized up to equivalence the metrics on Tn which are invariant by the induced transitive action of a connected closed subgroup of the group of Euclidean motions (they exist only in such dimensions and are pseudo-Riemannian of split type) and described explicitly their geodesics. In this short note we present the geometric meaning of the latter being null, time- or space-like. On the other hand, it is well-known that Tn is diffeomorphic to G(Hn), the space of all oriented geodesics of the n-dimensional hyperbolic space. For n = 3 and n = 7, we compute now a pseudo-Riemannian invariant of Tn (involving its periodic geodesics) that will be useful to show that Tn and G(Hn) are not isometrically equivalent, provided that the latter is endowed with any of the metrics which are invariant by the canonical action of the identity component of the isometry group of H.