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

TL;DR

This paper explores the relationship between conformally flat hypersurfaces in 4D space forms and 3-metrics with the Guichard condition, providing a method to generate conformally flat 3-metrics from constant curvature 2-metrics.

Contribution

It establishes a correspondence between conformally flat hypersurfaces and 3-metrics with the Guichard condition, and develops an evolution process linking 2-metrics of constant curvature to conformally flat 3-metrics.

Findings

An evolution of orthogonal 2-metrics with constant Gauss curvature -1 determines conformally flat 3-metrics.

A one-parameter family of conformally flat 3-metrics can be generated from certain 2-metrics.

The study provides a framework for understanding the geometry of conformally flat hypersurfaces.

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.