Classification of closed conformally flat Lorentzian manifolds with unipotent holonomy

TL;DR

This paper classifies closed, conformally flat Lorentzian manifolds with unipotent holonomy, revealing four geometric types and characterizing those with essential conformal flows, primarily homeomorphic to products of spheres and circles.

Contribution

It provides a comprehensive classification of such manifolds based on holonomy and geometric properties, including the characterization of manifolds admitting essential conformal flows.

Findings

Manifolds are either homeomorphic to S^{n-1} × S^1 or nilmanifolds of degree ≤ 3.

Classification into four geometric types based on the intersection with a holonomy-invariant isotropic flag.

Manifolds with essential conformal flows are of two types, both homeomorphic to S^{n-1} × S^1.

Abstract

We classify closed, conformally flat Lorentzian manifolds of dimension $n \geq 3$ with unipotent holonomy in PO(2,n). They are all Kleinian and fall into four different geometric types according to the intersection of the image of the developing map with a holonomy-invariant isotropic flag. They are homeomorphic to $S^{n-1} \times S^1$ or a nilmanifold of degree at most three, up to a finite cover. We classify those admitting an essential conformal flow; these fall into two geometric types, both homeomorphic to $S^{n-1} \times S^1$ up to finite cover.