A note on tractor bundles and codimension two spacelike immersions

We study conformal tractor bundles from an extrinsic viewpoint, relating them to codimension two spacelike immersions into Lorentzian manifolds. We show that, at least locally, every Riemannian conformal structure admits a natural realization of its normal conformal tractor bundle as the pullback of the tangent bundle of a suitably constructed Lorentzian ambient space. Finally, we reformulate the classical equations characterizing parallel sections of the normal conformal tractor bundle in this extrinsic setting, showing that they can be expressed entirely in terms of the geometry of the associated spacelike immersion. This extrinsic perspective provides additional geometric insight into parallel standard tractors and conformal holonomy.

💡 Research Summary

This paper revisits the theory of normal conformal tractor bundles from an extrinsic standpoint, showing that every Riemannian conformal structure can locally be realized as the pull‑back of the tangent bundle of a suitably constructed Lorentzian ambient space. The authors begin by recalling the intrinsic definition of a conformal tractor bundle: a rank‑(n+2) real vector bundle equipped with a Lorentzian metric, a distinguished light‑like line subbundle, and a compatible linear connection (the normal tractor connection) uniquely determined by the underlying conformal class for (n\ge 3).

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