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Standard tractors and the conformal ambient metric construction. (English) Zbl 1039.53021
A classical result of Elie Cartan states that every conformal manifold of dimension \(\geq 3\) admits a canonical normal Cartan connection. T. Y. Thomas, in the 1920s, introduced the notion of tractor bundles, which provide a description of Cartan bundles and Cartan connections in terms of linear connections on certain vector bundles. C. Fefferman and C. R. Graham, in 1985, gave a new point of view. They considered a pseudo-conformal structure of signature \((p,q)\) on a manifold \(M\) as a ray bundle \(S^2T^*M\supset {\mathcal Q}\to M\), and they associated to a conformal structure a pseudo-Riemannian metric of signature \((p+1,q+1)\) on \({\mathcal Q}\times (-1,1)\), the so-called ambient metric.
In the paper under review, the authors relate the ambient metric construction to the conformal standard tractor bundle and its canonical linear connection. They show that from any ambient metric that satisfies a weakening of the usual normalisation condition, one can construct the conformal standard tractor bundle and the normal standard tractor connection, which are equivalent to the Cartan bundle and the Cartan connection. They apply this result to obtain a procedure to get tractor formulae for all conformal invariants that can be obtained from the ambient metric construction. They also obtain information on ambient metrics which are Ricci flat to higher order than guaranteed by the results of Fefferman-Graham.

MSC:
53B15 Other connections
53A30 Conformal differential geometry (MSC2010)
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