NDLI: On Bäcklund and Ribaucour transformations for surfaces with constant negative curvature

Content Provider	Springer Nature Link
Author	Tenenblat, Keti Goulart, Claudia
Copyright Year	2015
Abstract	Given a surface in the Euclidean 3-space with constant negative Gaussian curvature, the composition of Bäcklund transformations generates a 4-parameter family of surfaces with the same curvature. Since Ribaucour transformations of such a surface also provides a 4-parameter family of surfaces of the same type, one has the following natural question. Are these two methods equivalent? The answer is negative in general. Necessary and sufficient conditions for the surfaces given by the two procedures to be congruent are obtained. An explicit example is given of a composition of Bäcklund transformations which is not a Ribaucour transformation. The analytic interpretation of this result provides a method of producing solutions of the sine-Gordon equation, distinct from those obtained by Bäcklund transformations. New exact solutions are given and some of the corresponding surfaces are visualized.
Ending Page	102
Page Count	20
Starting Page	83
File Format	PDF
ISSN	00465755
e-ISSN	15729168
Journal	Geometriae Dedicata
Issue Number	1
Volume Number	181
Language	English
Publisher	Springer Netherlands
Publisher Date	2015-09-22
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Geometry Bäcklund transformations Surfaces with constant negative curvature Nonlinear second-order hyperbolic equations Immersions (minimal, prescribed curvature, tight, etc.) Sine-Gordon equation Surfaces in Euclidean space Soliton-like equations Methods of Riemannian geometry, including PDE methods; curvature restrictions Ribaucour transformations
Content Type	Text
Resource Type	Article
Subject	Geometry and Topology

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