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

João Paulo dos Santos ; Keti Tenenblat

Symmetry, integrability and geometry: methods and applications, Tome 9 (2013)

Résumé

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.

Détail

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

Bibliographie

[1] Barbosa J. L., Ferreira W., Tenenblat K., “Submanifolds of constant sectional curvature in pseudo-Riemannian manifolds”, Ann. Global Anal. Geom., 14 (1996), 381–401 | DOI | MR | Zbl

[2] Cartan E., “La déformation des hypersurfaces dans l'espace conforme réel à {$n \ge 5$} dimensions”, Bull. Soc. Math. France, 45 (1917), 57–121 | MR | Zbl

[3] Corro A. V., Martínez A., Milán F., “Complete flat surfaces with two isolated singularities in hyperbolic 3-space”, J. Math. Anal. Appl., 366 (2010), 582–592, arXiv: 0905.2371 | DOI | MR | Zbl

[4] Ferreira W., Soluções invariantes pelos grupos de simetria de Lie das Equações Generalizadas Intrínsecas de Laplace e de sinh-Gordon elíptica e propriedades geométricas das subvariedades associadas, Ph. D. thesis, Universidade de Brasília, 1994 | Zbl

[5] Gálvez J. A., Martínez A., Milán F., “Flat surfaces in the hyperbolic $3$-space”, Math. Ann., 316 (2000), 419–435 | DOI | MR

[6] Guichard C., Sur les systèmes triplement indéterminés et sur les systèmes triplement orthogonaux, Gauthier-Villars, Paris, 1905 | Zbl

[7] Hertrich-Jeromin U., Introduction to Möbius differential geometry, London Mathematical Society Lecture Note Series, 300, Cambridge University Press, Cambridge, 2003 | DOI | MR | Zbl

[8] Hertrich-Jeromin U., “On conformally flat hypersurfaces and Guichard's nets”, Beiträge Algebra Geom., 35 (1994), 315–331 | MR | Zbl

[9] Hertrich-Jeromin U., Suyama Y., “Conformally flat hypersurfaces with Bianchi-type Guichard nets”, Osaka J. Math. (to appear)

[10] Hertrich-Jeromin U., Suyama Y., “Conformally flat hypersurfaces with cyclic Guichard net”, Internat. J. Math., 18 (2007), 301–329 | DOI | MR | Zbl

[11] Kokubu M., Rossman W., Saji K., Umehara M., Yamada K., “Singularities of flat fronts in hyperbolic space”, Pacific J. Math., 221 (2005), 303–351, arXiv: math.DG/0401110 | DOI | MR | Zbl

[12] Kokubu M., Umehara M., Yamada K., “Flat fronts in hyperbolic 3-space”, Pacific J. Math., 216 (2004), 149–175, arXiv: math.DG/0301224 | DOI | MR | Zbl

[13] Lafontaine J., “Conformal geometry from the Riemannian viewpoint”, Conformal Geometry (Bonn, 1985/1986), Aspects Math. E, 12, Vieweg, Braunschweig, 1988, 65–92 | MR

[14] Lamé G., Leçons sur les coordonnés curvilignes et leurs diverses applications, Mallet-Bachelier, Paris, 1859

[15] Lie S., Theorie der Transformationsgruppen, B. G. Teubner, Leipzig, 1888; 1890; 1893

[16] Martinez A., dos Santos J. P., Tenenblat K., “Helicoidal flat surfaces in the hyperbolic 3-space”, Pacific J. Math. (to appear)

[17] Olver P. J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, 107, Springer-Verlag, New York, 1986 | DOI | MR | Zbl

[18] Olver P. J., “Symmetry groups and group invariant solutions of partial differential equations”, J. Differential Geom., 14 (1979), 497–542 | MR | Zbl

[19] Rabelo M. L., Tenenblat K., “Submanifolds of constant nonpositive curvature”, Mat. Contemp., 1 (1991), 71–81 | MR | Zbl

[20] Suyama Y., “Conformally flat hypersurfaces in Euclidean 4-space”, Nagoya Math. J., 158 (2000), 1–42 | MR | Zbl

[21] Suyama Y., “Conformally flat hypersurfaces in Euclidean 4-space, II”, Osaka J. Math., 42 (2005), 573–598 | MR | Zbl

[22] Suyama Y., “Conformally flat hypersurfaces in Euclidean 4-space and a class of Riemannian 3-manifolds”, Sūrikaisekikenkyūsho Kōkyūroku, 2001, no. 1236, 60–89 | MR | Zbl

[23] Tenenblat K., Transformations of manifolds and applications to differential equations, Pitman Monographs and Surveys in Pure and Applied Mathematics, 93, Longman, Harlow, 1998 | MR | Zbl

[24] Tenenblat K., Winternitz P., “On the symmetry groups of the intrinsic generalized wave and sine-{G}ordon equations”, J. Math. Phys., 34 (1993), 3527–3542 | DOI | MR | Zbl