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

摘要

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.