  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.
