An Indefinite Kähler Metric on the Space of Oriented Lines

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 R with the tangent bundle of S. Thus, the round metric on S induces a Kähler structure on T which turns out to have a metric of neutral signature. It is shown that the isometry group of this metric is isomorphic to the isometry group of the Euclidean metric on R. The geodesics of this metric are either planes or helicoids in R. The signature of the metric induced on a surface Σ in T is determined by the degree of twisting of the associated line congruence in R, and we show that, for Σ Lagrangian, the metric is either Lorentz or totally null. For such surfaces it is proven that the Keller-Maslov index counts the number of isolated complex points of J inside a closed curve on Σ.