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.
Categories: math.DG math.GT