Abstract
We study higher dimensional versions of shearfree null-congurences in conformal Lorentzian manifolds. We show that such structures induce a subconformal structure and a partially integrable almost CR-structure on the leaf space and we classify the Lorentzian metrics that induce the same subconformal structure. In the last section we survey some known applications of the correspondence between almost CR-structures and shearfree null-congurences in dimension 4.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Burns, D. Jr., Diederich, K., Shnider, S.: Distinguished curves in pseudoconvex boundaries. Duke Math. J. 44(2), 407–431 (1977)
Čap, A., Gover, A.R.: CR-tractors and the Fefferman space. Indiana Univ. Math. J. 57(5), 2519–2570 (2008)
Debney, G.C., Kerr, R.P., Schild, A.: Solutions of the Einstein and Einstein-Maxwell equations. J. Math. Phys. 10, 1842–1854 (1969)
Fefferman, C.L.: Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains. Ann. Math. (2) 103(2), 395–416 (1976)
Hill, C.D., Lewandowski, J., Nurowski, P.: Einstein’s equations and the embedding of 3-dimensional CR manifolds. Indiana Univ. Math. J. 57(7), 3131–3176 (2008). https://doi.org/10.1512/iumj.2008.57.3473
Jacobowitz, H.: The canonical bundle and realizable CR hypersurfaces. Pac. J. Math. 127(1), 91–101 (1987)
Lee, J.M.: The Fefferman metric and pseudo-Hermitian invariants. Trans. Amer. Math. Soc. 296(1), 411–429 (1986)
Lewandowski, J., Nurowski, P., Tafel, J.: Einstein’s equations and realizability of CR manifolds. Class. Quantum Gravity 7(11), L241–L246 (1990)
Robinson, I.: Null electromagnetic fields. J. Math. Phys. 2, 290–291 (1961)
Robinson, I., Trautman, A.: Cauchy-Riemann structures in optical geometry. In: Proceedings of the Fourth Marcel Grossmann Meeting on General Relativity, Part A, B (Rome, 1985), pp. 317–324. North-Holland, Amsterdam (1986)
Acknowledgements
This project was supported by DP130103485 of the Australian Research Council; D.V.A. carried out this work at IITP and was supported by an RNF grant (project no.14-50-00150).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Alekseevsky, D.V., Ganji, M., Schmalz, G. (2018). CR-Geometry and Shearfree Lorentzian Geometry. In: Byun, J., Cho, H., Kim, S., Lee, KH., Park, JD. (eds) Geometric Complex Analysis. Springer Proceedings in Mathematics & Statistics, vol 246. Springer, Singapore. https://doi.org/10.1007/978-981-13-1672-2_2
Download citation
DOI: https://doi.org/10.1007/978-981-13-1672-2_2
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-1671-5
Online ISBN: 978-981-13-1672-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)