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There is a one-to-one correspondence between associated families of genericconformally flat (local-)hypersurfaces in 4-dimensional space forms andconformally flat 3-metrics with the Guichard condition. In this paper, we studythe space of conformally flat 3-metrics with the Guichard condition: for aconformally flat 3-metric with the Guichard condition in the interior of thespace, an evolution of orthogonal (local-) Riemannian $2$-metrics with constantGauss curvature $-1$ is determined; for a $2$-metric belonging to a certainclass of orthogonal analytic $2$-metrics with constant Gauss curvature $-1$, aone-parameter family of conformally flat 3-metrics with the Guichard conditionis determined as evolutions issuing from the $2$-metric.

Curvilinear coordinates on generic conformally flat hypersurfaces and constant curvature 2-metrics