The Ribaucour transformation in Lie sphere geometry

Burstall, Francis E.; Hertrich-Jeromin, Udo

doi:10.1016/j.difgeo.2006.04.007

Cited by 21 publications

(53 citation statements)

References 13 publications

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“…Integrability aspects of the surface theory in Lie geometry have been studied by Ferapontov [F1, F2], Musso and Nicolodi [MN], and Burstall and Hertrich-Jeromin [BHJ1,BHJ2].…”

Section: Introductionmentioning

confidence: 99%

On organizing principles of discrete differential geometry. Geometry of spheres

Bobenko¹,

Suris²

2007

Russ. Math. Surv.

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Abstract. Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the transformation group principle (smooth geometric objects and their discretizations are invariant with respect to the same transformation group) and the consistency principle (discretizations of smooth parametrized geometries can be extended to multidimensional consistent nets). The main concrete geometric problem discussed in this survey is a discretization of curvature line parametrized surfaces in Lie geometry. By systematically applying the Discretization Principles we find a discretization of curvature line parametrization which unifies the circular and conical nets.

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“…Integrability aspects of the surface theory in Lie geometry have been studied by Ferapontov [F1, F2], Musso and Nicolodi [MN], and Burstall and Hertrich-Jeromin [BHJ1,BHJ2].…”

Section: Introductionmentioning

confidence: 99%

On organizing principles of discrete differential geometry. Geometry of spheres

Bobenko¹,

Suris²

2007

Russ. Math. Surv.

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“…The following Bianchi k-cube theorem was proved in [11] for k = 3 in the context of triply orthogonal systems of Euclidean space. A nice proof in the setup of Lie sphere geometry was recently given in [2], where an indication was also provided of how the general case can be settled by using results of [12] for discrete orthogonal nets together with an induction argument. …”

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confidence: 99%

The vectorial Ribaucour transformation for submanifolds and applications

Dajczer

Florit

Tojeiro

2007

Trans. Amer. Math. Soc.

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Abstract. In this paper we develop the vectorial Ribaucour transformation for Euclidean submanifolds. We prove a general decomposition theorem showing that under appropriate conditions the composition of two or more vectorial Ribaucour transformations is again a vectorial Ribaucour transformation. An immediate consequence of this result is the classical permutability of Ribaucour transformations. Our main application is to provide an explicit local construction of an arbitrary Euclidean n-dimensional submanifold with flat normal bundle and codimension m by means of a commuting family of m Hessian matrices on an open subset of Euclidean space R n . Actually, this is a particular case of a more general result. Namely, we obtain a similar local construction of all Euclidean submanifolds carrying a parallel flat normal subbundle, in particular of all those that carry a parallel normal vector field. Finally, we describe all submanifolds carrying a Dupin principal curvature normal vector field with integrable conullity, a concept that has proven to be crucial in the study of reducibility of Dupin submanifolds.An explicit construction of all submanifolds with flat normal bundle of the Euclidean sphere carrying a holonomic net of curvature lines, that is, admitting principal coordinate systems, was given by Ferapontov in [8]. The author points out that his construction "resembles" the vectorial Ribaucour transformation for orthogonal systems developed in [13]. The latter provides a convenient framework for understanding the permutability properties of the classical Ribaucour transformation. This paper grew from an attempt to better understand the connection between those two subjects, as a means of unravelling the geometry behind Ferapontov's construction. This has led us to develop a vectorial Ribaucour transformation for Euclidean submanifolds, extending the transformation in [13] for orthogonal coordinate systems. It turns out that any n-dimensional submanifold with flat normal bundle of R n+m can be locally transformed by a suitable vectorial Ribaucour transformation to the inclusion map of an open subset of an n-dimensional subspace of R n+m . Inverting such a transformation yields the following explicit local construction of an arbitrary n-dimensional submanifold with flat normal bundle of R n+m by means of a commuting family of m Hessian matrices on an open subset of R n . Notice that carrying a principal coordinate system is not required.

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“…More recent results were given in [11] and in the context of Lie sphere geometry in [4]. Bianchi proved that a Permutability theorem also holds for Ribaucour transformations of surfaces with constant Gaussian or mean curvature.…”

Section: Theorem 32 Let M Be a Surface Of R 3 Without Umbilic Poinmentioning

confidence: 96%

“…We choose the Ribaucour constant to be C R = 1/2, and the constants of (4.12) to be A = B = 1, D = 0 then E = 5/4. It follows from the first case of Proposition 4.3, that f = sinh 2 x 1 + 5 4 2 cosh x 1 , g = 1 2…”

Section: New Exact Solutionsmentioning

confidence: 96%

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On Bäcklund and Ribaucour transformations for surfaces with constant negative curvature

Goulart

Tenenblat

2015

Geom Dedicata

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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.

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The Ribaucour transformation in Lie sphere geometry

Cited by 21 publications

References 13 publications

On organizing principles of discrete differential geometry. Geometry of spheres

On organizing principles of discrete differential geometry. Geometry of spheres

The vectorial Ribaucour transformation for submanifolds and applications

On Bäcklund and Ribaucour transformations for surfaces with constant negative curvature

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