| Summary: | <p>On a complex n-manifold with holomorphic projective connexion, any point has a neighbourhood U of which the space of geodesies has naturally the structure of a (Hausdorff) complex (2n-2)-manifold; it is shown that the complex structure of this auxiliary space encodes, in a sense, the original projective connexion by means of a complete analytic family of and#x2119;<sub>n-1</sub>'s. Rather strikingly, small deformations of the space of geodesies correspond precisely to small deformations of the projective connexion of the primary space.</p><p>Similar results are obtained concerning the space of null geodesies of a complex n-fold, n ≥ 4, with <em>conformal connexion</em>, a geometrical structure amounting to a holomorphic conformal structure plus a torsion tensor. For n=3 the analysis is complicated by the fact that the analytic family of submanifolds (quadrics; in this case and#x2119;<sub>1</sub>'s) representing the points of the primary space fails to be complete; but it can be completed to give a 4- dimensional family, effecting a unique embedding of the original 3-fold in a 4-fold with conformal structure, of which the conformal curvature is selfdual, in such a way that the induced conformal structure is the original one and such that the conformal torsion is related to the second conformal fundamental form of the hypersurface in a canonical linear fashion. In any case, the small deformations of the complex structure of the space of null geodesies correspond precisely to the small deformations of the conformal connexion. It is shown that a space of torsion-free null geodesies admits a holomorphic contact structure, and that conversely, for n ≥ 4, the admission of a contact structure forces the conformal torsion to vanish; for n=3, the contact form constructs automatically a unique metric on the ambient 4-fold in the previously constructed self-dual conformal class which solves Einstein's equations with cosmological constant 1 and blov/s up on the 3-fold, which is a general umbilic hypersurface. These results are in turn used to show that a real-analytic 3-fold with real-analytic positive definite conformal structure and a real-analytic symmetric form of conformal weight 1 can be embedded (in a locally unique fashion) in a real-analytic 4-fold with positive-definite conformal structure for which the conformal curvature is self-dual in such a way as to realize the given structures as the first and second conformal fundamental forms of the hypersurface; and it is shown that a real analytic 3-fold with positivedefinite conformal bounds a locally unique positive-definite solution of Einstein's equations with cosmological constant -1 as its umbilic conformal infinity. By contrast, these results fail when "real-analytic" is replaced by "smooth".</p>
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