A Lorenz surface (statistics) is one possible higher dimensional extension of a Lorenz curve proposed by Taguchi, Lunetta, Arnold, Koshevoy and Mosler. Analytic Expressions for Multivariate Lorenz Surfaces.
A Lorentz surface (geometry) is a two-dimensional oriented smooth manifold with a conformal equivalence class of Lorentzian metrics. It is the analogue of a Riemann surface in indefinite signature. Lorentz Surface.
Can the two definitions coincide? Is there a known example?
I think that if we purposely define a Lorenz surface to be some analytic function and construct an extension to 3-space via revolving the function about $y=x$ then we can obtain a surface. Then it would have to be proven that this surface has a conformal equivalence class of Lorentzian metrics.