Normal tractor conformal bundles and codimension two spacelike submanifolds in Lorentzian manifolds

TL;DR

This paper introduces a canonical construction of tractor conformal bundles for codimension two spacelike submanifolds in Lorentzian manifolds, linking their intrinsic and extrinsic geometries through normality conditions.

Contribution

It provides a new method to construct and analyze tractor conformal bundles associated with spacelike submanifolds, characterizing when these bundles are standard and normal.

Findings

Characterization of when the induced connection defines a tractor connection

Conditions under which the tractor conformal bundle is standard

Normality conditions relating intrinsic and extrinsic geometry

Abstract

For every codimension two spacelike submanifold of a Lorentz manifold and each choice of a normal lightlike vector field, we introduce a canonical way to construct a tractor conformal bundle. We characterize when the induced connection of a such submanifold defines a tractor connection and then, in this case, when this tractor conformal bundle with the induced connection is standard for the induced metric. Finally, the normality condition for this tractor conformal bundle endowed with the induced connection is characterized in terms of a strong relationship between the intrinsic and the extrinsic geometry of the starting spacelike submanifold.