Vectorial Ribaucour transformations for the Lamé equations

Liu, Q. P.; Mañas, Manuel

doi:10.1088/0305-4470/31/10/003

Cited by 12 publications

(17 citation statements)

References 8 publications

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“…The proof relies on a general decomposition theorem for the vectorial Ribaucour transformation for submanifolds (Theorem 14), according to which the composition of two or more vectorial Ribaucour transformations with appropriate conditions is again a vectorial Ribaucour transformation. The latter extends a similar result of [13] for the case of orthogonal systems and implies, in particular, the classical permutability of Ribaucour transformations for surfaces and, more generally, the permutability of vectorial Ribaucour transformations for submanifolds.…”

supporting

confidence: 80%

“…Finally, (13) implies that the exterior derivatives of both sides in the first equality of (14) coincide.…”

Section: When This Is the Case There Existsmentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

“…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 .…”

mentioning

confidence: 99%

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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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supporting

confidence: 80%

“…Finally, (13) implies that the exterior derivatives of both sides in the first equality of (14) coincide.…”

Section: When This Is the Case There Existsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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The vectorial Ribaucour transformation for submanifolds and applications

Dajczer

Florit

Tojeiro

2007

Trans. Amer. Math. Soc.

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“…19,1,20 We also show that for the reduced pseudo-circular case we get the generalization to this pseudo-Euclidean case of the discrete Ribaucour-Bianchi transformation. [20][21][22] We finally characterize the discrete fundamental transformations for the symmetric lattices of Ref. 18.…”

Section: Introductionmentioning

confidence: 99%

From integrable nets to integrable lattices

Mañas

2002

Journal of Mathematical Physics

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Inspired by the results of Jonas, Einsenhart, Demoulin, and Bianchi on the permutability property of classical geometrical transformations of conjugate nets and its reductions-of pseudo-orthogonal, pseudo-symmetric, and pseudo-Egorov typesdressing transformations of the N-component KP hierarchy ͑described within the Grassmannian͒ are used to generate quadrilateral lattices and its corresponding reductions. As a byproduct we get the corresponding discrete dressing transformations; in particular, we characterize the vectorial fundamental discrete transformations preserving the symmetric lattice.

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The Vectorial Ribaucour Transformation for Submanifolds of Constant Sectional Curvature

Guimarães

Tojeiro

2017

J Geom Anal

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We obtain a reduction of the vectorial Ribaucour transformation that preserves the class of submanifolds of constant sectional curvature of space forms, which we call the L-transformation. It allows to construct a family of such submanifolds starting with a given one and a vector-valued solution of a system of linear partial differential equations. We prove a decomposition theorem for the L-transformation, which is a far-reaching generalization of the classical permutability formula for the Ribaucour transformation of surfaces of constant curvature in Euclidean three space. As a consequence, we derive a Bianchi-cube theorem, which allows to produce, from k initial scalar L-transforms of a given submanifold of constant curvature, a whole k-dimensional cube all of whose remaining 2 k − (k + 1) vertices are submanifolds with the same constant sectional curvature given by explicit algebraic formulae. We also obtain further reductions, as well as corresponding decomposition and Bianchicube theorems, for the classes of n-dimensional flat Lagrangian submanifolds of C n and n-dimensional Lagrangian submanifolds with constant curvature c of the complex projective space CP n (4c) or the complex hyperbolic space CH n (4c) of complex dimension n and constant holomorphic curvature 4c.

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Vectorial Ribaucour transformations for the Lamé equations

Cited by 12 publications

References 8 publications

The vectorial Ribaucour transformation for submanifolds and applications

The vectorial Ribaucour transformation for submanifolds and applications

From integrable nets to integrable lattices

The Vectorial Ribaucour Transformation for Submanifolds of Constant Sectional Curvature

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