Séminaire de Géométrie
Cauchy-Riemann geometry and Einstein Lorentzian metricsArman Taghavi-Chabert (Université polytechnique de Lodz)
Friday 22 March 2024 13:30 - Tours - 1180 (Bât. E2)
Résumé :
This talk is concerned with two aspects of the interaction between Cauchy-Riemann geometry and Lorentzian conformal geometry. On the one hand, it was realised, notably through the work of Sir Roger Penrose and his associates, and that of the Warsaw group led by Andrzej Trautman, that CR three-manifolds underlie Einstein Lorentzian four-manifolds that admit non-shearing congruences of null geodesics. These foliations play a fundamental rôle in mathematical relativity, and constitute one of the original ingredients in the formulation of twistor theory.
On the other hand, motivated by his investigation of CR chains, Charles Fefferman in 1976 constructed, in a canonical way, a Lorentzian conformal structure on a circle bundle over a given strictly pseudoconvex Cauchy-Riemann (CR) manifolds of hypersurface type.
After reviewing these two independent developments, I will show how these can be related to each other, by presenting modifications of Fefferman’s original construction, where the conformal structure is "perturbed" by some semi-basic one-form, which encodes additional data on the CR three-manifold. Our setup allows us to reinterpret previous works by Lewandowski, Nurowski, Tafel, Hill, and independently, by Mason. Metrics in such a perturbed Fefferman conformal class whose Ricci tensor satisfies certain degeneracy conditions, are only defined off sections of the Fefferman bundle, which may be viewed as "conformal infinity". The prescriptions on the Ricci tensor can then be reduced to differential constraints on the CR three-manifold in terms of a "complex density" and the CR data of the perturbation one-form. One such constraint turns out to arise from a non-linear, or gauged, analogue of a second-order differential operator on densities. A solution thereof provides a criterion for the existence of a CR function and, under certain conditions, for CR embeddability. This talk is partly based on arxiv:2303.07328
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