Artículos científicos
Publicado 2009-08-31
Palabras clave
- Dressing actions,
- Grassmannian systems,
- Lorentzian isothemic surfaces
Cómo citar
Dussan, M. P., & Magid, M. A. (2009). A note about isothermic surfaces in Rn−j,j. Revista Integración, Temas De matemáticas, 26(2), 61–76. Recuperado a partir de https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/175
Resumen
In this note we survey our results on the description of ti-melike isothermic surfaces in Rn-j,j using the Grassmannian systems or U/K-systems. We give the natural extensions of the definition of Ribaucour and Darboux transformations for timelike isothermic surfaces and review how those transformations correspond to dressing actions of suitable simple elements.
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Referencias
[1] A.I. Bobenko, “Surfaces in terms of 2 by 2 matrices, old and new integrable cases”,Harmonic maps and Integrable systems, Edited by A.P. Fordy and J. Wood, (1994),83-127. Vieweg.
[2] M. Bruck, X. Du, J. Park, C-L. Terng, “The Submanifolds Geometries associatedto Grassmannian Systems”, Memoirs of A.M.S. 735 (2002).
[3] F. Burstall, “Isothermic surfaces: conformall geometry, Clifford algebras and Integrable systems”, Preprint, math-dg/0003096.
[4] F. Burstall, U. Hertrich-Jeromin, F. Pedit, U. Pinkall, “Curved flats and isothermic surfaces”, Math. Z. no. 2, 225 (1997).
[5] J. Ciesliski, P. Goldstein, A. Sym, “Isothermic surfaces in E3 as soliton surfaces”, Phys. Lett. A 205 (1995), 37-43.
[6] J. Ciesliski, “The Darboux-Bianchi transformation for isothermic surfaces”, Differential Geom. Appl. 7 (1997), 1-28.
[7] M. Dajczer, R. Tojeiro, “Commuting Codazzi tensors and the Ribaucour transformation
for submanifolds”, Results in Math. 44, (2003), 258-278.
[8] M.P. Dussan, M.A. Magid, “Timelike isothermic surfaces associated to Grassmannian systems”, Doc. Math. 10, (2005), 527-549.
[9] M.P. Dussan, M.A. Magid, “Complex Timelike isothermic surfaces and their Geometric transformations”, Balkan J. Geom. Appl. 11, (2006), no. 1, 39-53.
[10] A. Fujioka, J. Inoguchi, “Spacelike surfaces and Harmonic Inverse Mean curvature”, J. Math. Sci. Univ. Tokyo. 7, (2000). 657-698.
[11] U. Hertrich-Jeromin, F. Pedit, “Remarks on the Darboux transform of isothermic surfaces”, Doc. Math. 2, (1997), 313-333.
[12] M.A. Magid, “Lorenztian Isothermic surfaces in Rn-j ”, Rocky Mountain J.M. 35, (2005), 627-640.
[13] A. Pressley, G.B. Segal, Loop Groups. Oxford Science Publ. Clarendon Press, Oxford. (1986).
[14] C.L. Terng, “Soliton equations and Differential Geometry”, J. Differential Geom. 45, (1997), no. 2. 407-445.
[15] C.L. Terng, K. Uhlenbeck, “Backlund transformations and loop group actions”, Comm. Pure Appl. Math. 53, (2000), 1-75.
[16] C. Tian, “Bäcklund transformation on surfaces with K = −1 in R2,1”, J. of Geom. and Phys. 22 (1997), 212-218.
[17] D. Zuo, Q. Chen, Y. Cheng, “Gp,qm,n-System II and diagonalizable timelike immersions in Rp,m ”, Inverse Problems, 20 (2004), 319-329.
[2] M. Bruck, X. Du, J. Park, C-L. Terng, “The Submanifolds Geometries associatedto Grassmannian Systems”, Memoirs of A.M.S. 735 (2002).
[3] F. Burstall, “Isothermic surfaces: conformall geometry, Clifford algebras and Integrable systems”, Preprint, math-dg/0003096.
[4] F. Burstall, U. Hertrich-Jeromin, F. Pedit, U. Pinkall, “Curved flats and isothermic surfaces”, Math. Z. no. 2, 225 (1997).
[5] J. Ciesliski, P. Goldstein, A. Sym, “Isothermic surfaces in E3 as soliton surfaces”, Phys. Lett. A 205 (1995), 37-43.
[6] J. Ciesliski, “The Darboux-Bianchi transformation for isothermic surfaces”, Differential Geom. Appl. 7 (1997), 1-28.
[7] M. Dajczer, R. Tojeiro, “Commuting Codazzi tensors and the Ribaucour transformation
for submanifolds”, Results in Math. 44, (2003), 258-278.
[8] M.P. Dussan, M.A. Magid, “Timelike isothermic surfaces associated to Grassmannian systems”, Doc. Math. 10, (2005), 527-549.
[9] M.P. Dussan, M.A. Magid, “Complex Timelike isothermic surfaces and their Geometric transformations”, Balkan J. Geom. Appl. 11, (2006), no. 1, 39-53.
[10] A. Fujioka, J. Inoguchi, “Spacelike surfaces and Harmonic Inverse Mean curvature”, J. Math. Sci. Univ. Tokyo. 7, (2000). 657-698.
[11] U. Hertrich-Jeromin, F. Pedit, “Remarks on the Darboux transform of isothermic surfaces”, Doc. Math. 2, (1997), 313-333.
[12] M.A. Magid, “Lorenztian Isothermic surfaces in Rn-j ”, Rocky Mountain J.M. 35, (2005), 627-640.
[13] A. Pressley, G.B. Segal, Loop Groups. Oxford Science Publ. Clarendon Press, Oxford. (1986).
[14] C.L. Terng, “Soliton equations and Differential Geometry”, J. Differential Geom. 45, (1997), no. 2. 407-445.
[15] C.L. Terng, K. Uhlenbeck, “Backlund transformations and loop group actions”, Comm. Pure Appl. Math. 53, (2000), 1-75.
[16] C. Tian, “Bäcklund transformation on surfaces with K = −1 in R2,1”, J. of Geom. and Phys. 22 (1997), 212-218.
[17] D. Zuo, Q. Chen, Y. Cheng, “Gp,qm,n-System II and diagonalizable timelike immersions in Rp,m ”, Inverse Problems, 20 (2004), 319-329.