The Symmetry Group of Lam\'e's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces

TL;DR

This paper analyzes the symmetry group of Lamé's system related to conformally flat hypersurfaces, revealing invariant solutions, geometric properties of Guichard nets, and introducing new classes of hypersurfaces using elliptic functions.

Contribution

It characterizes the symmetry group of Lamé's system satisfying Guichard conditions and constructs invariant solutions leading to new conformally flat hypersurfaces.

Findings

Symmetry group includes translations and dilations.

Guichard nets have constant Gaussian curvature and are foliated by flat, constant mean curvature surfaces.

New solutions involve Jacobi elliptic functions and correspond to novel hypersurfaces.

Abstract

We consider conformally flat hypersurfaces in four dimensional space forms with their associated Guichard nets and Lam\'e's system of equations. We show that the symmetry group of the Lam\'e's system, satisfying Guichard condition, is given by translations and dilations in the independent variables and dilations in the dependents variables. We obtain the solutions which are invariant under the action of the 2-dimensional subgroups of the symmetry group. For the solutions which are invariant under translations, we obtain the corresponding conformally flat hypersurfaces and we describe the corresponding Guichard nets. We show that the coordinate surfaces of the Guichard nets have constant Gaussian curvature, and the sum of the three curvatures is equal to zero. Moreover, the Guichard nets are foliated by flat surfaces with constant mean curvature. We prove that there are solutions of the Lam\'e's system, given in terms of Jacobi elliptic functions, which are invariant under translations, that correspond to a new class of conformally flat hypersurfaces.