manuscripta mathematica

, Volume 118, Issue 2, pp 181–189

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

Authors

Article

DOI: 10.1007/s00229-005-0576-z

Cite this article as:
Salvai, M. manuscripta math. (2005) 118: 181. doi:10.1007/s00229-005-0576-z

Abstract

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)

53B3053B3553C2253C3022F3032M1032Q15

Copyright information

© Springer-Verlag Berlin Heidelberg 2005