A concept of degenerate Bäcklund transformation is introduced for two-dimensional surfaces in many-dimensional Euclidean spaces. It is shown that if a surface in \( {\mathbb{R}}_n \) , n ≥ 4, admits a degenerate Bäcklund transformation, then this surface is pseudospherical, i.e., its Gauss curvature is constant and negative. The complete classification of the pseudospherical surfaces in \( {\mathbb{R}}_n \) , n ≥ 4, admitting the degenerate Bianchi transformations is proposed. Moreover, we also obtain a complete classification of the pseudospherical surfaces in \( {\mathbb{R}}_n \) , n ≥ 4, that admit degenerate Bäcklund transformations into straight lines.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Yu. A. Aminov and A. Sym, “On Bianchi and BŁcklund transformations of two-dimensional surfaces in E 4 ,” Math. Phys., Anal., Geom., 3, 75–89 (2000).
Yu. A. Aminov and A. Sym, “On Bianchi and BŁcklund transformations of two dimensional surfaces in four-dimensional Euclidean space,” in: Bäcklund and Darboux Transformations. The Geometry of Solitons, AARMS-CRM Workshop, June 4–9, 1999, Halifax, N.S., Canada, American Mathematical Society, Providence, RI (2001), pp. 91–93.
Yu. A. Aminov, Geometry of Submanifolds [in Russian], Naukova Dumka, Kiev (2002).
A. A. Borisenko and Yu. A. Nikolaevskii, “Grassmann manifolds and a Grassmann image of submanifolds,” Usp. Mat. Nauk, 46, No. 2, 41–83 (1991).
V. Gorkavyy, “On pseudospherical congruences in E 4 ,” Mat. Fiz., Anal., Geometr., 10, No. 4, 498–504 (2003).
V. A. Gor’kavyi, “Bianchi congruences of two-dimensional surfaces in E 4 ,” Mat. Sb., 196, No. 10, 79–102 (2005).
V. O. Gorkavyy and O. M. Nevmerzhytska, “Ruled surfaces as pseudospherical congruences,” Zh. Mat. Fiz.. Anal., Geometr., 5, No. 3, 359–374 (2009).
V. A. Gor’kavyi and E. N. Nevmerzhitskaya, “Two-dimensional pseudospherical surfaces with degenerate Bianchi transformation,” Ukr. Mat. Zh., 63, No. 11, 1460–1468 (2011); English translation: Ukr. Math. J., 63, No. 11, 1660–1669 (2012).
V. Gorkavyy, “An example of Bianchi transformation in E 4 ,” Zh. Mat. Fiz.. Anal., Geometr., 8, No. 3, 240–247 (2012).
V. A. Gor’kavyi, “Generalization of the Bianchi–Bäcklund transformation of pseudospherical surfaces,” in: Advances in Science and Technology. Contemporary Mathematics and Its Applications. Thematic Reviews [in Russian], 126 (2014), pp. 191–218.
É. G. Poznyak, “Geometric interpretation of regular solutions of the equation Z xy = sin Z,” Differents. Uravn., 15, No. 7, 1332–1336 (1979).
É. G. Poznyak and A. G. Popov, “Geometry of the sine-Gordon equation,” in: Advances in Science and Technology. Geometry [in Russian], 23 (1991), pp. 99–130.
A. G. Popov, Lobachevskii Geometry and Mathematical Physics [in Russian], Moscow University, Moscow (2012).
K. Tenenblat, Transformations of Manifolds and Applications to Differential Equations, Longman, London (1998).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, No. 1, pp. 38–51, January, 2016.
Rights and permissions
About this article
Cite this article
Gor’kavyi, V.A., Nevmerzhitskaya, E.N. Degenerate Bäcklund Transformation. Ukr Math J 68, 41–56 (2016). https://doi.org/10.1007/s11253-016-1207-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-016-1207-4