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.
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Salvai, M. On the geometry of the space of oriented lines of Euclidean space. manuscripta math. 118, 181–189 (2005). https://doi.org/10.1007/s00229-005-0576-z
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DOI: https://doi.org/10.1007/s00229-005-0576-z