Geodesics of the space of oriented lines of the Euclidean space

SALVAI, MARCOS

La Falda, Córdoba

Congreso; Egeo 2005; 2005

For n = 3 or n = 7 let L^n be the space of oriented lines in R^n. In a previous article we characterized up to equivalence the metrics on L^n 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 L^n is diffeomorphic to G(H^n), 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 L^n (involving its periodic geodesics) that will be useful to show that T^n and G( H^n) 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.