An Indefinite Kaehler Metric on the Space of Oriented Lines

Details An Indefinite Kaehler Metric on the Space of Oriented Lines An Indefinite Kaehler Metric on the Space of Oriented Lines

2004-07-28

arxiv.org/abs/math/0407490v1

J. London Math. Soc. 72 (2005) 497-509.

Mathematics

Differential Geometry

12 pages, AMS-LATEX

Scientific paper

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 $\Sigma$.

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