An Embedding Theorem for tractor bundles, and an application in conformal pseudo-Riemannian geometry

TL;DR

This paper extends the Gromov--Zimmer Embedding Theorem to tractor bundles with invariant connections and applies it to establish rigidity results for conformal group actions on certain pseudo-Riemannian manifolds.

Contribution

It generalizes the embedding theorem to a broader class of tractor bundles and demonstrates its application in conformal pseudo-Riemannian geometry.

Findings

Extended embedding theorem to tractor bundles with invariant connections

Proved rigidity of conformal actions of pseudo-unitary groups

Applied the theorem to closed, simply connected pseudo-Riemannian manifolds

Abstract

We provide an extension of the Gromov--Zimmer Embedding Theorem for Cartan geometries of [3] to tractor bundles carrying any invariant connection, including tractor connections and prolongation connections of first BGG operators for parabolic geometries. As an application, we prove a rigidity result for conformal actions of special pseudo-unitary groups on closed, simply connected, analytic pseudo-Riemannian manifolds.