A note on tractor bundles and codimension two spacelike immersions

TL;DR

This paper explores the relationship between conformal tractor bundles and codimension two spacelike immersions into Lorentzian manifolds, providing a new extrinsic perspective that links conformal geometry with Lorentzian ambient spaces.

Contribution

It introduces a method to realize normal conformal tractor bundles as pullbacks from Lorentzian ambient spaces and reformulates classical equations in this extrinsic setting.

Findings

Normal conformal tractor bundles can be realized as pullbacks from Lorentzian ambient spaces.

Classical equations for parallel tractors are reformulated in terms of spacelike immersion geometry.

Provides geometric insight into conformal holonomy and tractor parallelism.

Abstract

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.