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 isometry group of this metric is isomorphic to 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 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 $\mathbb{J}$ inside a closed curve on $\Sigma$.

12 pages, AMS-LATEX

References in corpus (2)

• On the Space of Oriented Affine Lines in $R^3$
• Einstein Metrics Adapted to Contact Structures on 3-Manifolds

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