The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces
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 vari...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2013 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2013
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149203 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces / J.P. dos Santos, K. Tenenblat // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 24 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862588702837964800 |
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| author | dos Santos, J.P. Tenenblat, K. |
| author_facet | dos Santos, J.P. Tenenblat, K. |
| citation_txt | The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces / J.P. dos Santos, K. Tenenblat // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 24 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.
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| first_indexed | 2025-11-27T02:28:05Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-149203 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-27T02:28:05Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | dos Santos, J.P. Tenenblat, K. 2019-02-19T18:34:43Z 2019-02-19T18:34:43Z 2013 The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces / J.P. dos Santos, K. Tenenblat // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 24 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53A35; 53C42 DOI: http://dx.doi.org/10.3842/SIGMA.2013.033 https://nasplib.isofts.kiev.ua/handle/123456789/149203 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. This paper is a contribution to the Special Issue “Symmetries of Dif ferential Equations: Frames, Invariants
 and Applications”. The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html.
 The authors were partially supported by CAPES/PROCAD and CNPq. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces Article published earlier |
| spellingShingle | The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces dos Santos, J.P. Tenenblat, K. |
| title | The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces |
| title_full | The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces |
| title_fullStr | The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces |
| title_full_unstemmed | The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces |
| title_short | The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces |
| title_sort | symmetry group of lamé's system and the associated guichard nets for conformally flat hypersurfaces |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149203 |
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