An indefinite Kaehler metric on the space of oriented lines.

Abstract

The total space of the tangent bundle of a Kähler manifold admits a canonical Kähler structure. Parallel translation identifies the space ${\mathbb{T}}$ of oriented affine lines in ${\mathbb{R}}^3$ with the tangent bundle of $S^2$. Thus the round metric on $S^2$ induces a Kähler structure on ${\mathbb{T}}$ 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 ${\mathbb{R}}^3$. The geodesics of this metric are either planes or helicoids in ${\mathbb{R}}^3$. The signature of the metric induced on a surface $\Sigma$ in ${\mathbb{T}}$ is determined by the degree of twisting of the associated line congruence in ${\mathbb{R}}^3$, and it is shown that, for $\Sigma$ Lagrangian, the metric is either Lorentz or totally null. For such surfaces it is proved that the Keller–Maslov index counts the number of isolated complex points of ${\mathbb{J}}$ inside a closed curve on $\Sigma$.