Research and Innovation Articles
Published 2009-08-31
Keywords
- Dressing actions,
- Grassmannian systems,
- Lorentzian isothemic surfaces
How to Cite
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. Retrieved from https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/175
Abstract
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.
Downloads
Download data is not yet available.
References
[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.