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A note on tractor bundles and codimension two spacelike immersions

Rodrigo Morón

TL;DR

The article develops an extrinsic framework for conformal tractor bundles by embedding an $n$-dimensional Riemannian conformal structure into a Lorentzian ambient space using codimension-two spacelike immersions. It constructs a 1-parameter family of ambient Lorentzian spaces and shows that, locally, the normal conformal tractor bundle arises as the pullback of the ambient tangent bundle when the ambient Ricci tensor vanishes along $r=0$, with a local normalization given by $\dot{\gamma}(0)=2P^g$. The paper then reformulates the equations for parallel standard tractors entirely in terms of the immersion geometry, establishing equivalence with the intrinsic tractor description and highlighting the role of the Schouten tensor and Ricci-flatness along the immersion. This extrinsic perspective provides new geometric insight into conformal holonomy and the structure of parallel tractors, while guaranteeing local realizability of the extrinsic model for any conformal structure.

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.

A note on tractor bundles and codimension two spacelike immersions

TL;DR

The article develops an extrinsic framework for conformal tractor bundles by embedding an -dimensional Riemannian conformal structure into a Lorentzian ambient space using codimension-two spacelike immersions. It constructs a 1-parameter family of ambient Lorentzian spaces and shows that, locally, the normal conformal tractor bundle arises as the pullback of the ambient tangent bundle when the ambient Ricci tensor vanishes along , with a local normalization given by . The paper then reformulates the equations for parallel standard tractors entirely in terms of the immersion geometry, establishing equivalence with the intrinsic tractor description and highlighting the role of the Schouten tensor and Ricci-flatness along the immersion. This extrinsic perspective provides new geometric insight into conformal holonomy and the structure of parallel tractors, while guaranteeing local realizability of the extrinsic model for any conformal structure.

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.
Paper Structure (6 sections, 5 theorems, 49 equations)

This paper contains 6 sections, 5 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on submanifold theory
  3. Conformal Tractor bundles and codimension two spacelike immersions
  4. A family of Lorentzian manifolds associated with a conformal structure
  5. Realization of the normal conformal tractor bundle via a class of codimension two spacelike immersions
  6. Parallel sections of the normal conformal tractor bundle

Key Result

Proposition 3.4

Let $\bigl(\Psi^{*}(T\widetilde{M}), \mathrm{Span}(\xi), \widetilde{g}, \widetilde{\nabla}\bigr)$ be the tractor bundle associated with the pair $(\Psi, \xi)$. Then $\bigl(\Psi^{*}(T\widetilde{M}), \mathrm{Span}(\xi), \widetilde{g}, \widetilde{\nabla}\bigr)$ is the normal conformal tractor bundle if for every $V,W\in \mathfrak{X}(M)$.

Theorems & Definitions (16)

  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • Remark 3.5
  • Remark 4.1
  • Theorem 5.1
  • Corollary 5.2
  • Remark 5.3
  • Remark 5.4
  • ...and 6 more