An Indefinite Kaehler Metric on the Space of Oriented Lines

Abstract

The total space of the tangent bundle of a K\"ahler manifold admits a canonical K\"ahler structure. Parallel translation identifies the space ${\Bbb{T}}$ of oriented affine lines in ${\Bbb{R}}^3$ with the tangent bundle of $S^2$. Thus, the round metric on $S^2$ induces a K\"ahler structure on ${\Bbb{T}}$ 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 ${\Bbb{R}}^3$. The geodesics of this metric are either planes or helicoids in ${\Bbb{R}}^3$. The signature of the metric induced on a surface $\Sigma$ in ${\Bbb{T}}$ is determined by the degree of twisting of the associated line congruence in ${\Bbb{R}}^3$, and we show that, for $\Sigma$ 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 ${\Bbb{J}}$ inside a closed curve on $\Sigma$.