Riv. Mat. Univ. Parma, Vol. 11, No. 1, 2020

Elisabetta Barletta [a] and Sorin Dragomir [a]

Robinson-Sparling construction of CR structures associated to shearfree null geodesic congruences

Pages: 9-68
Accepted in revised form: 7 May 2019
Mathematics Subject Classification (2010): 32V20, 32V30, 53C50, 53D10, 53Z05.
Keywords: CR structure, tangential Cauchy-Riemann equations, Fefferman metric, flag structure, optical structure, Lorentzian metric, space-time, Maxwell field, Dirac equation.
[a]: Università degli Studi della Basilicata, Dipartimento di Matematica, Informatica ed Economia, Via dell'Ateneo Lucano 10, Potenza, 85100, Italy

Abstract: We review the construction of Lorentzian metrics, such as Fefferman type metrics, associated to a given $$3$$-dimensional nondegenerate CR manifold $$M$$, and admitting shearfree null geodesic congruences $$N$$. This class of metrics is obtained by a lifting procedure from $$M$$ to $$M \times {\mathbb R}$$ devised by I. Robinson and A. Trautman (cf. [71]-[72]) and notably radiative gravitational fields are searched for (cf. e.g. R.K. Sachs, [74]) within the class. Conversely, nondegenerate CR structures arise (by the Robinson-Trautmann construction, [71]) on leaf spaces $${\mathfrak M}/N$$ associated to space-times $$\mathfrak M$$ adapted to given optical structures $$\big( (K, L), \, J)$$. The Graham-Sparling construction (cf. [40], [77]) is shown to be a particular case of Robinson-Trautman construction where the complex structure on the complex line bundle $${\rm Ker} (L) /K \to {\mathfrak M}$$ is induced by an $$f$$-structure with two complemented frames obtained as a covariant derivative of the given null Killing vector field $$N$$.

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