TOWARD A CLASSIFICATION OF CLOSED CONFORMALLY FLAT LORENTZIAN 3-MANIFOLDS WITH NILPOTENT HOLONOMY

Abstract

A conformally flat Lorentzian $3$-manifold determines a $(G, X)$ manifold structure, where $X$ is a conformally flat Lorentzian model space denoted by $\mathrm{Ein}^3$, and $G$ is $\mathrm{PO}(3, 2)$, the conformal group of $\mathrm{Ein}^3$. We find there are $9$ conjugacy classes of maximal nilpotent subgroups of $G$, which reduces the classification problem for closed $\mathrm{Ein}^3$ manifolds with nilpotent holonomy to $9$ cases. The case of unipotent holonomy was classified by Lee. We study a case that is represented by the internal direct product of the pointwise stabilizer of two disjoint photons and a unipotent group which pointwise stabilizes a photon that intersects the first two. We make progress toward a complete classification of this case. We make a full classification of the subcases such that the corresponding developing image $\Omega$ is either $\mathrm{Ein}^3$, or the complement of the two disjoint photons, or one of three subsets of the complement of the three photons. In these subcases, such a manifold admits either $S^1 \times S^2$ or the $3$-torus as a covering space, and except when $\Omega$ equals the complement of the three photons, the associated $(\mathrm{Stab}_G(\Omega), \Omega)$ structure is complete. Addressing a remark by Frances, in the final chapter we argue there is no connected sum operation for conformally flat Lorentzian manifolds.