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

Abstract

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.