A pseudo-spherical surface in R^4 does not admit two different Bianchi transformations
DOI:
https://doi.org/10.15673/2072-9812.1/2015.50150Keywords:
pseudo-spherical surface, Bianchi transformation, horocyclic coordinates, conjugate netAbstract
It is proved that if a pseudo-spherical surface in four-dimensional Euclidean space R4, which does not belong to R3, admits a Bianchi transformation, then this Bianchi transformation is unique.References
Aminov Yu.A., SymA., On Bianchi and Backlund transformations of two-dimensional surfaces in E4 // Math. Phys., An., Geom. – 2000. – V. 3. – P. 75-89.
Aminov Yu.A., Sym A., On Bianchi and Backlund transformations of two dimensional surfaces in four dimensional Euclidean space // Backlund and Darboux transformations. The geometry of solitons. AARMS-CRM Workshop, Halifax, NS, 1999, Canada, June 4-9. – Amer. Math. Soc. – 2001. – P. 91-93.
Aminov Yu.A., Geometry of Submanifolds. – Gordon & Breach Science Publ., Amsterdam. – 2001.
Борисенко А.А., Николаевский Ю.А., Многообразие Грассмана и грассманов образ подмногообразий // Успехи математических наук. – 1991. – Т. 46, № 2. – С. 41-83.
Cieslinski J., Goldstein P., Sym A., Isothermic surfaces in E3 as soliton surfaces // Phys. Lett., A. – 1995. – V. 205, N.1. – P. 37-43.
Gorkavyy V., On pseudo-spherical congruencies in E4 // Математическая физика, анализ, геометрия. – 2003. – Т. 10, В. 4. – С. 498-504.
Горькавый В.А., Конгруэнции Бьянки двумерных поверхностей в Е4 // Матем. сборник. – 2005. - Т. 196, № 10. – С. 79-102.
Gorkavyy V.O, Nevmerzhytska O.M., Ruled surfaces as pseudo-spherical congruencies // Журнал математической физики, анализа, геометрии. – 2009. – Т. 5, В. 3. – С. 359-374.
Горькавый В.А., Невмержицкая Е.Н., Аналог преобразования Бианки для двумерных поверхностей в пространстве S 3 x R1// Ма¬т. за¬ме¬т¬ки. – 2011. – Т. 89, № 6. – С. 833-845.
Gorkavyy V., Nevmerzhitska O., Pseudo-spherical submanifolds with degenerate Bianchi Transformation // Results in Mathematics. – 2011. – V. 60, No. 1. – P. 103-116.
Gorkavyy V., An example of Bianchi transformation in E4 // Журнал мат.ематической физики, анализа, геометрии. – 2012. – Т. 8, В. 3. – С. 240-247.
Горькавый В.А., Обобщение преобразования Бианки-Бэклунда псевдосферических поверхностей // Итоги науки и техники. Современная математика и ее приложения. Тематические обзоры. – 2014. – Т. 126. – С. 191-218.
Tenenblat K., Transformations of manifolds and applications to differential equations. – Pitman Monographs and Surveys in Pure Appl. Math, V. 93, Longman Sci. Techn., Harlow, Essex; Wiley, New York. – 1998.
Tenenblat K., Terng C.-L., Backlund's theorem for n-dimensional submanifolds of R2n-1 // Annals of Mathematics. – 1980. – V. 111. – P. 477-490.
