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Conformal circles and parametrizations of curves in conformal manifolds. (English) Zbl 0684.53016

The authors consider regular, nowhere isotropic curves \(\gamma\) in a Riemannian or pseudo-Riemannian manifold M the metric of which is defined up to a conformal change \(g\mapsto \Omega^ 2g\). They define conformal circles by a certain ordinary differential equation of third order which is shown to split into two separate conformally invariant equations. One of these equations defines a conformally invariant family of natural parameters, defining a projective structure on \(\gamma\) ; such a structure has been found by E. Cartan already, it can be defined for any nowhere isotropic curves \(\gamma\). The other equation singles out the conformal circles. By the help of these the authors construct conformal circle coordinates. Generalizations to the complex case and the relation to the twistor construction are mentioned.
Reviewer: R.Sulanke

MSC:

53A30 Conformal differential geometry (MSC2010)
53B20 Local Riemannian geometry
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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References:

[1] É. Cartan, Les espaces à connexion conforme, Oeuvres Complètes III. 1, Gauthiers-Villars, Paris, 1955, pp. 747-797.
[2] H. Friedrich and B. G. Schmidt, Conformal geodesics in general relativity, Proc. Roy. Soc. London Ser. A 414 (1987), no. 1846, 171 – 195. · Zbl 0629.53063
[3] R. Graham, A conformal normal form, (to appear).
[4] Roger Penrose and Wolfgang Rindler, Spinors and space-time. Vol. 2, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1986. Spinor and twistor methods in space-time geometry. · Zbl 0591.53002
[5] K. Yano, The theory of the Lie derivative and its applications, North Holland, Amsterdam, 1955.
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