There is a one-to-one correspondence between associated families of generic conformally flat (local-)hypersurfaces in 4-dimensional space forms and conformally flat 3-metrics with the Guichard condition. In this paper, we study the space of conformally flat 3-metrics with the Guichard condition: for a conformally flat 3-metric with the Guichard condition in the interior of the space, an evolution of orthogonal (local-)Riemannian 2-metrics with constant Gauss curvature $-1$ is determined; for a 2-metric belonging to a certain class of orthogonal analytic 2-metrics with constant Gauss curvature $-1$, a one-parameter family of conformally flat 3-metrics with the Guichard condition is determined as evolutions issuing from the 2-metric.
Francis E. BURSTALL. Udo HERTRICH-JEROMIN. Yoshihiko SUYAMA. "Curvilinear coordinates on generic conformally flat hypersurfaces and constant curvature 2-metrics." J. Math. Soc. Japan 70 (2) 617 - 649, April, 2018. https://doi.org/10.2969/jmsj/07027420