Conformally flat structures via hyperbolic geometry

TL;DR

This paper extends the duality between geometric structures related to hyperbolic space from surfaces to higher dimensions, providing new equations and a proof of the Weyl-Schouten theorem.

Contribution

It generalizes the duality and equations for conformally flat structures in hyperbolic geometry to arbitrary dimensions and proves a key theorem using this framework.

Findings

Established a duality between tensor pairs in higher dimensions.

Identified equations corresponding to the Gauss-Codazzi equations in higher dimensions.

Provided a hyperbolic geometry-based proof of the Weyl-Schouten theorem.

Abstract

A pair of tensors $(g,B)$ form the induced metric and shape operator of an immersion into hyperbolic space if and only if they satisfy the Gauss-Codazzi equations. Such a pair of tensors induce a pair $(\hat{g},\hat{B})$ related to the ideal boundary of hyperbolic space. Krasnov and Schlenker, and Bridgeman and Bromberg show in the surface case that there is a duality between $(g,B)$ and $(\hat{g},\hat{B})$. Moreover, $(g,B)$ solves the Gauss-Codazzi equations if and only if $(\hat{g},\hat{B})$ solve a corresponding set of equations. We show a similar duality exists and identify these corresponding equations for an arbitrary dimension, as well as show there exists a unique solution for $\hat{B}$ provided $\hat{g}$ is locally conformally flat. As an application, we offer a proof of the Weyl-Schouten theorem concerning locally conformally flat metrics that factors through hyperbolic geometry.