Robinson manifolds and the Chern-Robinson connection

The almost Robinson manifolds are even dimensional Lorentzian manifolds endowed with a complex subbundle of the complexify tangent bundle which is maximally totally null with respect to the complexified metric. These manifolds can be viewed as Lorentzian analogues of almost Hermitian manifolds. This geometric structure appears in the context of the general relativity for some solutions of Einstein equations like gravitational waves or black holes (Kerr or Taub-NUT black holes). The above formal definition is quite recent and due to Nurowski and Trautman ([15]). The interest focused on the Robinson manifolds has increased in the last years. Among the works on the subject, the following [1], [2], [9], [10], [12], [16], [19], [20] deal with relationships between Robinson manifolds and CR manifolds and also about classification of almost Robinson manifolds. For instance, in [10], the authors give a classification of almost Robinson manifolds based on the concept of intrinsic torsion. In this article, we continue to investigate the geometry of almost Robinson manifolds (started in [18]) by means of the Witt structures. Recall that a Witt structure on the (complexify) tangent bundle of a pseudo-Riemannian manifold is a decomposition into totally null subbundles (directly inspired by the Witt decomposition of pseudo-euclidean spaces). As described in ( [10], [18]), the maximally totally null complex subbundle of an almost Robinson manifold gives rise to a Witt decomposition on the complexify tangent bundle of the almost Robinson manifold. Also it is possible to equiped the (complexify) tangent bundle together with a metric connection preserving the Witt structure (note that a such connection necessarily possesses torsion). The so called Lichnerowicz connection introduced in ([18]) provides a such example, however, there is no canonical choice for this type of connections. In this article, we introduce an other metric connection preserving the Witt structure called the Chern-Robinson connection with the property to coincide with the ∂-operator associated to the almost Robinson structure (which is a counterpart of the Chern connection in Hermitian geometry ( [6], [11])). Now, by means of this connection together with its curvature and its torsion, we can investigate the geometry of almost Robinson manifolds from a new perspective. In that follows, we restrict this investigation mainly to so called almost Fefferman-Robinson manifolds (subclass of almost Robinson manifolds realized as line bundles over CR strictly pseudoconvex manifolds). The plan of this article is as follow. In section 2, the definitions of almost optical and almost Robinson manifolds are reviewed thus examples. We also describe the Witt structure on their complexify tangent bundle. Section 3 is devoted to construct the Chern-Robinson connection for an almost Robinson manifold and to derive first Bianchi identity for the curvature of the connection of almost Fefferman-Robinson manifolds (Propositions 3.1 and 3.2). In the last part of this section, we define various type of curvatures, especially, we define the Chern-Robinson curvature tensor and the Chern-Moser-Robinson tensor (which is the counterpart of the Chern-Moser tensor [7] for CR strictly pseudoconvex manifolds). The Section 4 is devoted to prove (Proposition 4.2 and Corollary 4.2) that the Chern-Moser-Robinson tensor is an optical conformal invariant for the strongly geodetic (in the sense of Definition 2.3 after) almost Fefferman-Robinson manifolds. In Section 5, we derive a second Bianchi identity for the Chern-Robinson curvature tensor and the Chern-Moser-Robinson tensor of a Fefferman-Robinson manifold (Propositions 5.2 and 5.3). As application of the second Bianchi identity, we give a formula expressing explicitly the Chern-Moser-Robinson tensor of an Einstein-Fefferman-Robinson manifold with parallel torsion in terms of the metric and the torsion (Proposition 5.4). This formula is, in particular, valid for a locally symmetric Fefferman-Robinson manifold Einstein-Fefferman (Corollary 5.1) and is similar to those already obtained in [17] for contact locally subsymmetric spaces.

Most of definitions given in this section can be found in [1], [2], [9], [10], [15], [16].

In the following, (M 2m+2 , g) is a (2m + 2)-dimensional Lorentzian manifold. Definition 2.1 An almost optical structure (cf. [9]) on (M 2m+2 , g) is a null line subbundle R of T M .

An almost optical structure gives rise to a filtration and a grading of T M :

The vector bundle S := R ⊥ /R is called screen bundle associated to R. Note that the Lorentzian metric g induces a riemannian metric g S on S.

Definition 2.2 An almost Robinson (or complex almost optical) structure (cf. [10], [15], [16]) on (M 2m+2 , g) is a rank-(m + 1) complex subbundle N of T C M which is totally null with respect to the complexified metric g C (i.e N ⊥ = N ).

In this case, N ∩N is a complex null line subbundle of T C M and there exists a re

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