An Indefinite Kähler Metric on the Space of Oriented Lines

@article{Guilfoyle2004AnIK,
  title={An Indefinite K{\"a}hler Metric on the Space of Oriented Lines},
  author={Brendan Guilfoyle and Wilhelm Klingenberg},
  journal={Journal of the London Mathematical Society},
  year={2004},
  volume={72}
}
The total space of the tangent bundle of a Kähler manifold admits a canonical Kähler structure. Parallel translation identifies the space T of oriented affine lines in R3 with the tangent bundle of S2. Thus the round metric on S2 induces a Kähler structure on T which turns out to have a metric of neutral signature. It is shown that the identity component of the isometry group of this metric is isomorphic to the identity component of the isometry group of the Euclidean metric on R3. 

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