Null-geodesics in complex conformal manifolds and the LeBrun correspondence

Abstract

In the complex-Riemannian framework we show that a conformal manifold containing a compact, simply-connected, null-geodesic is conformally¯at. In dimension 3 we use the LeBrun correspondence, that views a conformal 3-manifold as the conformal in®nity of a selfdual four-manifold. We also ®nd a relation between the conformal invariants of the conformal in®nity and its ambient.Our main result (section 4, Theorem 4) states that, if a conformal complex n-manifold admits a rational curve as a null-geodesic, then it is conformally¯at (see also [17] for Partially supported by the SFB 288 of the DFG.Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 6/1/15 10:56 AM the case of a complex projective manifold). The proof uses the properties of Jacobi ®elds along the considered compact, simply-connected, null-geodesic: namely, we compute the normal bundle of a compact, simply-connected, null-geodesic, and we show that the small deformations of the latter as a compact curve, or as a null-geodesic, coincide (section 4, Proposition 5). In addition to that, we use, for the (more di½cult) case of dimension 3, a criterion for conformal¯atness from [2], and we apply it to a locally de®ned, by the LeBrun correspondence (see below), self-dual ambient.The other topic of this paper uses implicitly another application of twistor theory: It has been shown by LeBrun [8], [9], that any conformal 3-manifold can be locally realized as the conformal in®nity of a self-dual Einstein (with non-zero scalar curvature) 4-manifold. We have, thus, a local correspondence assigning to a conformal structure in dimension 3 a self-dual Einstein metric in dimension 4, which we call the LeBrun correspondence.As conformal structures of both manifolds are encoded in the complex, resp. CR, structure of their twistor spaces, they are implicitly related, for example if the 3-manifold M is conformally¯at, its ambient space N equally is. It is, however, di½cult to obtain an explicit relation between the conformal invariants of M and those of N by twistorial methods, as there is no simple expression of the Cotton-York tensor of M 3 in twistorial terms, and the twistorial interpretation of the Weyl tensor of N 4 is highly non-linear [2].In this paper we ®nd a relation between these two conformal invariants of the manifolds involved in the LeBrun correspondence, or, more generally, of an umbilic submanifold M 3 and of its self-dual ambient space N 4 . It appears that the Weyl tensor of N 4 identically vanishes along M 3 , and thus the Cotton-York tensor of N 4 , restricted to M 3 , is conformally invariant and can be identi®ed with the Cotton-York tensor of M 3 ; in this case, it is also equal to the normal derivative of the Weyl tensor of N 4 (section 3, Theorem 1). This gives conditions for an open self-dual 4-manifold to admit a conformal in®nity.The paper is organized as follows: in section 2 we recall a few basic facts about complex-Riemannian and -conformal geometry, in section 3 we relate the conformal invaria...

Homogénéité locale pour les métriques riemanniennes holomorphes en dimension 3

“…Il s'agit d'un fibré principal au-dessus de M de groupe structural D r (C n ), le groupe des r-jets en 0 de biholomorphismes locaux de C n qui préservent 0. Par exemple, le fibré des 1-repères s'identifie avec l'ensemble des bases de tous les espaces tangents à M et [3].…”

On the Weyl tensor of a self-dual complex 4-manifold

“…(In fact, all we need to define the conformal structure c on the 4-manifold M is just the holomorphic bundle L 2 ; in odd dimensions the situation is different; see [2]. )…”

“…As a consequence, Jacobi fields on γ induce particular local sections in N (γ), which turn out to be (conformally invariant) solutions of a differential operator of order 2 on N (γ); see [2].…”

“…If we assume, in addition, that the compact null-geodesic is simply-connected, then the above result can be deduced, using Theorem 2 , for any self-dual 4-manifold, and also (using the LeBrun correspondence [12]) for the case of a conformal complex 3-manifold [2]; In fact, we have recently proven [2] that the existence of a compact, simply-connected, null-geodesic on an n-dimensional conformal complex manifold implies its conformal flatness, for any n ≥ 3 (different methods are used for n > 3).…”

“…This method is used in [2] to prove the same thing starting from a conformal complex 3-manifold (using the LeBrun correspondence, i.e. the local realization of a conformal 3-manifold as the conformal infinity of a (germ-unique) self-dual manifold [12]), but we also show there, by different methods, that, in all generality, a conformal complex n-manifold (n ≥ 4) containing a compact, simply-connected null-geodesic is conformally flat ( [2], Theorem 4).…”