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

TL;DR

This work generalizes the Gromov–Zimmer Embedding Theorem to tractor bundles carrying invariant connections on Cartan geometries, enabling a unified treatment of parallel tractor sections and their algebraic symmetry data. By combining the tractor-embedding framework with the Borel density and Frobenius-type arguments, the authors derive sharp algebraic constraints on how simple groups can act conformally on pseudo-Riemannian manifolds, culminating in a rigidity result for actions by groups locally isomorphic to $SU(p',q')$. In the conformal setting, the embedding yields a bound $p \ge 2p' - 1$ on the ambient signature and, in the extremal case, forces a transition to a conformally flat geometry modeled on the Möbius space $\mathbf{M}^{p,q}$; analytic and simply connected hypotheses lead to a precise classification as the universal cover of $\mathbf{M}^{p,q}$. The approach blends parabolic-geometry machinery (BGG operators, prolongation connections, and tractor calculus) with dynamical tools (balanced isotropy, local flows) to connect tractor-parallel solutions to geometric rigidity and global structure.

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.

Theorems & Definitions (44)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Proposition 2.1
• proof
• Definition 2.3
• Theorem 2.2: Corollary 3.2 in bfm.zimemb of the Borel Density Theorem
• Theorem 3.1
• proof