Conformally flat Lorentz manifolds and Fefferman Lorentz metrics

TL;DR

This paper explores the geometry of conformally flat Lorentz manifolds with Fefferman structures, establishing key theorems linking curvature tensors and conformal transformations in these specialized manifolds.

Contribution

It introduces Fefferman-Lorentz structures on manifolds and proves theorems relating curvature vanishing and conformal rigidity in compact Fefferman-Lorentz manifolds.

Findings

Vanishing of Weyl conformal curvature tensor coincides with Chern-Moser curvature tensor.

Established an analogue of the conformal rigidity theorem for certain Fefferman-Lorentz manifolds.

Demonstrated the role of causal conformal vector fields in these geometric structures.

Abstract

We study conformal Fefferman-Lorentz manifolds introduced by Fefferman. To do so, we introduce Fefferman-Lorentz structure on (2n+2)-dimensional manifolds. By using causal conformal vector fields preserving that structure, we shall establish two theorems on compact Fefferman-Lorentz manifolds: One is the coincidence of vanishing curvature between Weyl conformal curvature tensor of Fefferman metrics on a Lorentz manifold $S^1\times N$ and Chern-Moser curvature tensor on a strictly pseudoconvex CR-manifold $N$. Another is the analogue of the conformal rigidity theorem of Obata and Lelong to the compact Fefferman-Lorentz manifolds admitting noncompact closed causal conformal Fefferman-Lorentz transformations.