Ribaucour Transformation Research Articles

Relatives Geometries

In this paper we consider M a fixed hypersurface in Euclidean space and we introduce two types of spaces relative to M, of type I and type II. We observe that when M is a hyperplane, the two geometries coincides with the isotropic geometry. By applying the theory to a Dupin hypersurface M, we define a relative Dupin hypersurface M of type I and type II , we provide necessary and sufficient conditions for a relative hypersurface M to be relative Dupin parameterized by relative lines of curvature, in both spaces. Moreover, we provides a relationship between the Dupin hypersurfaces locally associated to M by a Ribaucour transformation and the type II Dupin hypersurfaces relative M. We provide explicit examples of the Dupin hypersurface relative to a hyperplane, torus, S1 x Rn-1 and S2 x Rn-2, in both spaces.

Conformally flat submanifolds with flat normal bundle

We prove that any conformally flat submanifold with flat normal bundle in a conformally flat Riemannian manifold is locally holonomic, that is, admits a principal coordinate system. As one of the consequences of this fact, it is shown that the Ribaucour transformation can be used to construct an associated large family of immersions with induced conformally flat metrics holonomic with respect to the same coordinate system.

Ribaucour coordinates

We discuss results for the Ribaucour transformation of curves or of higher dimensional smooth and discrete submanifolds. In particular, a result for the reduction of the ambient dimension of a submanifold is proved and the notion of Ribaucour coordinates is derived using a Bianchi permutability theorem. Further, we discuss smoothing of semi-discrete curvature line nets and an interpolation by Ribaucour transformations.

On Bäcklund and Ribaucour transformations for hyperbolic linear Weingarten surfaces

We consider Bäcklund transformations for hyperbolic linear Weingarten surfaces in Euclidean 3-space. The composition of these transformations is obtained in the Permutability Theorem that generates a 4-parameter family of surfaces of the same type. Since a Ribaucour transformation of a hyperbolic linear Weingarten surface also gives a 4-parameter family of such surfaces, one has the following natural question. Are these two methods equivalent, as it occurs with surfaces of constant positive Gaussian curvature or constant mean curvature? By obtaining necessary and sucient conditions for the surfaces given by the two procedures to be congruent.The analytic interpretation of the geometric results is given in terms of solutions of the sine-Gordon equation.

The Ribaucour Transformation for Hypersurfaces of Two Space Forms and Conformally Flat Hypersurfaces

We develop a Ribaucour transformation for the class of conformally flat hypersurfaces $$f:M^{3} \rightarrow \mathbb {Q}_s^{4}(c)$$ with three distinct principal curvatures of a pseudo-Riemannian space form of dimension 4, constant curvature c and index $$s\in \{0, 1\}$$ , as well as for the class of hypersurfaces $$f:M^{3} \rightarrow \mathbb {Q}_s^{4}(c)$$ with three distinct principal curvatures for which there exists another isometric immersion $$\tilde{f}:M^{3} \rightarrow \mathbb {Q}^{4}_{\tilde{s}}(\tilde{c})$$ with $$\tilde{c}\ne c$$ . It gives a process to produce a family of new elements of those classes starting from a given one and a solution of a linear system of PDE’s. This enables us to construct explicit new examples of hypersurfaces in both classes.

The Vectorial Ribaucour Transformation for Submanifolds of Constant Sectional Curvature

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 Bianchi-cube theorems, for the classes of n-dimensional flat Lagrangian submanifolds of $${\mathbb {C}}^n$$ and n-dimensional Lagrangian submanifolds with constant curvature c of the complex projective space $${\mathbb {C}}{\mathbb {P}}^n(4c)$$ or the complex hyperbolic space $${\mathbb {C}}{\mathbb {H}}^n(4c)$$ of complex dimension n and constant holomorphic curvature 4c.

Hypersurfaces of two space forms and conformally flat hypersurfaces

We address the problem of determining the hypersurfaces $f\colon M^{n} \to \mathbb{Q}_s^{n+1}(c)$ with dimension $n\geq 3$ of a pseudo-Riemannian space form of dimension $n+1$, constant curvature $c$ and index $s\in \{0, 1\}$ for which there exists another isometric immersion $\tilde{f}\colon M^{n} \to \mathbb{Q}^{n+1}_{\tilde s}(\tilde{c})$ with $\tilde{c}\neq c$. For $n\geq 4$, we provide a complete solution by extending results for $s=0=\tilde s$ by do Carmo and Dajczer and by Dajczer and the second author. Our main results are for the most interesting case $n=3$, and these are new even in the Riemannian case $s=0=\tilde s$. In particular, we characterize the solutions that have dimension $n=3$ and three distinct principal curvatures. We show that these are closely related to conformally flat hypersurfaces of $\mathbb{Q}_s^{4}(c)$ with three distinct principal curvatures, and we obtain a similar characterization of the latter that improves a theorem by Hertrich-Jeromin. We also derive a Ribaucour transformation for both classes of hypersurfaces, which gives a process to produce a family of new elements of those classes, starting from a given one, in terms of solutions of a linear system of PDE's. This enables us to construct explicit examples of three-dimensional solutions of the problem, as well as new explicit examples of three-dimensional conformally flat hypersurfaces that have three distinct principal curvatures.

On Bäcklund and Ribaucour transformations for surfaces with constant negative curvature

Given a surface in the Euclidean 3-space with constant negative Gaussian curvature, the composition of Backlund 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 Backlund 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 Backlund transformations. New exact solutions are given and some of the corresponding surfaces are visualized.

A note on Ribaucour transformations in Lie sphere geometry

Following Burstall and Hertrich-Jeromin we study the Ribaucour transformation of Legendre submanifolds in Lie sphere geometry. We give an explicit parametrization of the resulted Legendre submanifold $\hat{F}$ of a Ribaucour transformation, via a single real function $\tau$ which represents the regular Ribaucour sphere congruence $s$ enveloped by the original Legendre submanifold $F$.

A connection between flat fronts in hyperbolic space and minimal surfaces in Euclidean space

A geometric construction is provided that associates to a given flat front in \(\mathbb {H}^3\) a pair of minimal surfaces in \(\mathbb {R}^3\) which are related by a Ribaucour transformation. This construction is generalized associating to a given frontal in \(\mathbb {H}^3\), a pair of frontals in \(\mathbb {R}^3\) that are envelopes of a smooth congruence of spheres. The theory of Ribaucour transformations for minimal surfaces is reformulated in terms of a complex Riccati ordinary differential equation for a holomorphic function. This enables one to simplify and extend the classical theory, that in principle only works for umbilic free and simply connected surfaces, to surfaces with umbilic points and non-trivial topology. Explicit examples are included.

Ribaucour transformations for isometric immersions of pseudo-Riemannian space forms

In this paper, we give the explicit construction for isometric immersions of pseudo-Riemannian space forms via loop group actions. As a consequence, we obtain a Ribaucour transformation and its permutability theorem for isometric immersions of pseudo-Riemannian space forms, which is a generalization of the permutability theorem for surfaces in R 3 , and a family of new isometric immersions of the pseudo-Riemannian space forms from a given one via a purely algebraic algorithm.

Transformations of Flat Lagrangian Immersions and Egoroff Nets

We associate a natural $\lambda$-family ($\lambda \in \R \setminus \{0\} $) of flat Lagrangian immersions in $\C^n$ with non-degenerate normal bundle to any given one. We prove that the structure equations for such immersions admit the same Lax pair as the first order integrable system associated to the symmetric space $\frac{\U(n) \ltimes \C^n}{\OO(n) \ltimes \R^n}$. An interesting observation is that the family degenerates to an Egoroff net on $\R^n$ when $\lambda \to 0$. We construct an action of a rational loop group on such immersions by identifying its generators and computing their dressing actions. The action of the generator with one simple pole gives the geometric Ribaucour transformation and we provide the permutability formula for such transformations. The action of the generator with two poles and the action of a rational loop in the translation subgroup produce new transformations. The corresponding results for flat Lagrangian submanifolds in $\C P^{n-1}$ and $\p$-invariant Egoroff nets follow nicely via a spherical restriction and Hopf fibration.

The vectorial Ribaucour transformation for submanifolds and applications

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 n -dimensional submanifold with flat normal bundle and codimension m m by means of a commuting family of m m Hessian matrices on an open subset of Euclidean space R n \mathbb {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.

The Ribaucour transformation in Lie sphere geometry

We discuss the Ribaucour transformation of Legendre (contact) maps in its natural context: Lie sphere geometry. We give a simple conceptual proof of Bianchi's original Permutability Theorem and its generalisation by Dajczer–Tojeiro as well as a higher dimensional version with the combinatorics of a cube. We also show how these theorems descend to the corresponding results for submanifolds in space forms.

Ribaucour Transformations for Hypersurfaces in Space Forms

The theory of Ribaucour transformations for hypersurfaces in space forms is established. For any such hypersurface M, that admits orthonormal principal vector fields, it was shown the existence of a totally umbilic hypersurface locally associated to M by a Ribaucour transformation. A method of obtaining linear Weingarten surfaces in a three-dimensional space form is provided. By applying the theory, a new one-parameter family of complete constant mean curvature (cmc) surfaces in the unit sphere, locally associated to the flat torus, is obtained. The family contains a class of complete cmc cylinders in the sphere. In particular, one gets a family of complete minimal surfaces and minimal cylinders, locally associated to the Clifford torus.

LOOP GROUP ACTIONS AND THE RIBAUCOUR TRANSFORMATIONS FOR FLAT LAGRANGIAN SUBMANIFOLDS

The Ribaucour transformations for flat Lagrangian submanifolds in Cn and CPn via loop group actions are given. As a consequence, the authors obtain a family of new flat Lagrangian submanifolds from a given one via a purely algebraic algorithm. At the same time, it is shown that such Ribaucour transformation always comes with a permutability formula.

Ribaucour transformations for constant mean curvature and linear Weingarten surfaces

We provide a method to obtain linear Weingarten surfaces from a given such surface, by imposing a one parameter algebraic condition on a Ribaucour transformation. Our main result extends classical results for surfaces of constant Gaussian or mean curvature. By applying the theory to the cylinder, we obtain a two-parameter family of complete linear Weingarten surfaces (hyperbolic, elliptic and tubular), asymptotically close to the cylinder, which have constant mean curvature when one of the parameters vanishes. The family contains n-bubble Weingarten surfaces which are 1-periodic, have genus zero and two ends of geometric index m, where n/m is an irreducible rational number. Their total curvature vanishes, while the total absolute curvature is 8πn. We also apply the method to obtain families of complete constant mean curvature surfaces, associated to the Delaunay surfaces, which are 1-periodic for special values of the parameter.

Commuting Codazzi tensors and the Ribaucour transformation for submanifolds

In this paper we develop a general theory for the Ribaucour transformation of sub-manifolds. In the process, a beautiful correspondence between Ribaucour transforms of a submanifold and Codazzi tensors that commute with its second fundamental form is revealed.

Minimal Surfaces Obtained by Ribaucour Transformations

We consider Ribaucour transformations between minimal surfaces and we relate such transformations to generating planar embedded ends. Applying Ribaucour transformations to Enneper's surface and to the catenoid, we obtain new families of complete, minimal surfaces, of genus zero, immersed in R3, with infinitely many embedded planar ends or with any finite number of such ends. Moreover, each surface has one or two nonplanar ends. A particular family is obtained from the catenoid, for each pair (n,m), n≠m, such that nm0 is an irreducible rational number. For any such pair, we get a 1-parameter family of finite total curvature, complete minimal surfaces with n+2 ends, n embedded planar ends and two nonplanar ends of geometric index m, whose total curvature is −4π(n+m). The analytic interpretation of a Ribaucour transformation as a Backlund type transformation and a superposition formula for the nonlinear differential equation ΔΦ= e-2Φ is included.

On Ribaucour transformations and applications to linear Weingarten surfaces

We present a revised definition of a Ribaucour transformation for submanifolds of space forms, with flat normal bundle, motivated by the classical definition and by more recent extensions. The new definition provides a precise treatment of the geometric aspect of such transformations preserving lines of curvature and it can be applied to submanifolds whose principal curvatures have multiplicity bigger than one. Ribaucour transformations are applied as a method of obtaining linear Weingarten surfaces contained in Euclidean space, from a given such surface. Examples are included for minimal surfaces, constant mean curvature and linear Weingarten surfaces. The examples show the existence of complete hyperbolic linear Weingarten surfaces in Euclidean space.