Conformal actions of nilpotent groups on pseudo-Riemannian manifolds

TL;DR

This paper investigates how connected nilpotent Lie groups can act conformally on compact pseudo-Riemannian manifolds, establishing bounds on the nilpotence degree and characterizing manifolds when this bound is achieved.

Contribution

It provides a bound on the nilpotence degree of groups acting conformally and characterizes the manifolds when this bound is attained, using Cartan geometry techniques.

Findings

Nilpotence degree of conformal actions is at most 2p+1.

Maximal nilpotence degree implies the manifold is conformally equivalent to a universal model.

Uses Cartan geometry to prove bounds and characterizations.

Abstract

We study conformal actions of connected nilpotent Lie groups on compact pseudo-Riemannian manifolds. We prove that if a type-(p,q) compact manifold M supports a conformal action of a connected nilpotent group H, then the degree of nilpotence of H is at most 2p+1, assuming p <= q; further, if this maximal degree is attained, then M is conformally equivalent to the universal type-(p,q), compact, conformally flat space, up to finite covers. The proofs make use of the canonical Cartan geometry associated to a pseudo-Riemannian conformal structure.