Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 033, 27 pages      arXiv:1304.4694      https://doi.org/10.3842/SIGMA.2013.033
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

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

João Paulo dos Santos and Keti Tenenblat
Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília-DF, Brazil

Received October 01, 2012, in final form April 12, 2013; Published online April 17, 2013

Abstract
We consider conformally flat hypersurfaces in four dimensional space forms with their associated Guichard nets and Lamé's system of equations. We show that the symmetry group of the Lamé'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 Lamé's system, given in terms of Jacobi elliptic functions, which are invariant under translations, that correspond to a new class of conformally flat hypersurfaces.

Key words: conformally flat hypersurfaces; symmetry group; Lamé's system; Guichard nets.

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