Higher dimensional conformal metrics from PDEs and null-surface formulation of GR

We analyze the relationship between n-dimensional conformal metrics and a certain class of partial differential equations (PDEs) that are in duality with the eikonal equation. In particular, we extend the null-surface formulation of general relativity (GR) to higher dimensions and give explicit expressions for the components of the metric and the generalized Wuenschmann-like metricity conditions. We also compute the equation that the conformal factor must satisfy in order the metric to be a solution of the Einstein equations. (paper)