# An Indefinite Kähler Metric on the Space of Oriented Lines

```@article{Guilfoyle2004AnIK,
title={An Indefinite K{\"a}hler Metric on the Space of Oriented Lines},
author={Brendan Guilfoyle and Wilhelm Klingenberg},
journal={Journal of the London Mathematical Society},
year={2004},
volume={72}
}```
• Published 28 July 2004
• Mathematics
• Journal of the London Mathematical Society
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 R3 with the tangent bundle of S2. Thus the round metric on S2 induces a Kähler structure on 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 R3.
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## References

SHOWING 1-10 OF 24 REFERENCES
Monopoles and geodesics
AbstractUsing the holomorphic geometry of the space of straight lines in Euclidean 3-space, it is shown that every static monopole of chargek may be constructed canonically from an algebraic curve by
Einstein Metrics Adapted to Contact Structures on 3-Manifolds
Mathematics Department, Institute of Technology Tralee, Tralee, Co. Kerry, Ireland.(February 8, 2008)The Newman-Penrose-Perjes formalism is applied to smooth contact structures on riemannian
On the Space of Oriented Affine Lines in \$R^3\$
• Mathematics
• 2004
We introduce a local coordinate description for the correspondence between the space of oriented affine lines in Euclidean \${\Bbb{R}}^3\$ and the tangent bundle to the 2-sphere. These can be utilised
On the space of oriented affine lines in \$\$ \mathbb{R}^3 \$\$
• Mathematics
• 2004
Abstract.We introduce a local coordinate description for the correspondence between the space of oriented affine lines in Euclidean \$\$ \mathbb{R}^3 \$\$ and the tangent bundle to the 2-sphere.
Spinors and space-time
• Physics
• 1984
In the two volumes that comprise this work Roger Penrose and Wolfgang Rindler introduce the calculus of 2-spinors and the theory of twistors, and discuss in detail how these powerful and elegant
Partial Differential Equations of Mathematical Physics
Preface Introduction 1. The Classical Equations 2. Applications of the integral theorems of green and stokes 3. Two-dimensional problems 4. Conformal Representation 5. Equations in three variables 6.
Symplectic Geometry
These are lecture notes for two courses, taught at the University of Toronto in Spring 1998 and in Fall 2000. Our main sources have been the books " Symplectic Techniques " by Guillemin-Sternberg and
Partial Differential Equations of Mathematical Physics
THE main work of mathematical physicists is to represent the sequence of phenomena in time and space by means of differential equations, and to solve these equations. Even the revolution effected by
Contact 3-manifolds twenty years since J. Martinet's work
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