manuscripta mathematica

, Volume 118, Issue 2, pp 181–189

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


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


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 

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.FaMAF - CIEMCiudad UniversitariaCórdobaArgentina

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