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An indefinite Kähler metric on the space of oriented lines

By Brendan Guilfoyle and Wilhelm Klingenberg


Abstract. 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 3 with the tangent bundle of S 2. Thus, the round metric on S 2 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 3. The geodesics of this metric are either planes or helicoids in R 3. 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 3, 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. 1

Year: 2005
DOI identifier: 10.1112/s0024610705006605
OAI identifier: oai:CiteSeerX.psu:
Provided by: CiteSeerX
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