Conformal actions of nilpotent groups on pseudo-Riemannian manifolds

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 $p\le q$; 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 or cyclic covers. The proofs make use of the canonical Cartan geometry associated to a pseudo-Riemannian conformal structure