Angel Ferrández

Geometrical particle models on 3D null curves

Abstract The simplest (2+1)-dimensional mechanical systems associated with light-like curves, already studied by Nersessian and Ramos, are reconsidered. The action is linear in the curvature of the particle path and the moduli spaces of solutions are completely exhibited in 3-dimensional Minkowski background, even when the action is not proportional to the pseudo-arc length of the trajectory.

Null generalized helices in Lorentz–Minkowski spaces

We obtain a Lancret-type theorem for null generalized helices in Lorentz– Minkowski spaces L n .I nL 3 we find that the only null generalized helices are the ordinary null helices. However, in L 5 we have to consider two types of null generalized helices according to whether the axis is non-null or null. In both cases we obtain the solutions to the natural equations problem.

A conformal variational approach for helices in nature

We propose a two step variational principle to describe helical structures in nature. The first one is governed by an energy action which is a linear function in both curvature and torsion allowing to describe nonclosed structures including elliptical, spherical, and conical helices. These appear as rhumb lines in right cylinders constructed over plane curves. The model is completed with a conformal alternative which, in particular, gives a description of closed structures. The energy action is linear in the curvatures when computed in a conformal spherical metric. Now, helices appear as making a constant angle with a Villarceau flow and so they are loxodromes in surfaces which are stereographic projections of Hopf tubes, in particular, anchor rings, revolution tori, and Dupin cyclides. The model satisfies the requirements of simplicity and beauty as reflected in the three main principles that head its construction: least action, topological, and quantization. According to the latter, the main entities an...

Willmore tori and Willmore-Chen submanifolds in pseudo-Riemannian spaces

Abstract We exhibit a new method to find Willmore tori and Willmore-Chen submanifolds in spaces endowed with pseudo-Riemannian warped product metrics, whose fibres are homogeneous spaces. The chief points are the invariance of the involved variational problems with respect to the conformal changes of the metrics on the ambient spaces and the principle of symmetric criticality. They allow us to relate the variational problems with that of generalized elastic curves in the conformal structure of the base space. Among other applications we get a rational one-parameter family of Willmore tori in the standard anti De Sitter 3-space shaped on an associated family of closed free elastic curves in the once punctured standard 2-sphere. We also obtain rational one-parameter families of Willmore-Chen submanifolds in standard pseudo-hyperbolic spaces. As an application of a general approach to our method, we give nice examples of pseudo-Riemannian 3-spaces which are foliated with leaves being non-trivial Willmore tori. More precisely, the leaves of this foliation are Willmore tori which are conformal to non-zero constant mean curvature flat tori.

Submanifolds in pseudo-Euclidean spaces satisfying the condition Δx=Ax+B

In this paper we study pseudo-Riemannian submanifolds in ℝn+k/tsatisfying the condition Δx=Ax+B, whereA is an endomorphism of ℝn+k/tandB is a constant vector in ℝn+k/t. We give a characterization theorem whenA is a self-adjoint endomorphism. As for hypersurfaces we are able to obtain a classification theorem for any endomorphismA.

Relativistic particles with rigidity and torsion in D =3 spacetimes

Models describing relativistic particles, where Lagrangian densities depend linearly on both the curvature and the torsion of the trajectories, are revisited in D = 3 Lorentzian spacetimes with constant curvature. The moduli spaces of trajectories are completely and explicitly determined. Trajectories are Lancret curves including ordinary helices. To get the geometric integration of the solutions, we design algorithms that essentially involve the Lancret program as well as the notions of scrolls and Hopf tubes. The most interesting and consistent models appear in anti-de Sitter spaces, where the Hopf mappings, both the standard and the Lorentzian ones, play an important role. The moduli subspaces of closed solitons in anti-de Sitter settings are also obtained. Our main tool is the isoperimetric inequality in the hyperbolic plane. The mass spectra of these models are also obtained. The main characteristic of the anti-de Sitter space comes from the presence of real gravity, which becomes essential to find some system with only massive states. This fact, on one hand, has no equivalent in flat spaces, where spectra necessarily present tachyonic sectors and, on the other hand, solves an early stated problem.

s-Degenerate curves in Lorentzian space forms

In this paper we introduce s-degenerate curves in Lorentzian space forms as those ones whose derivative of order s is a null vector provided that s>1 and all derivatives of order less than s are space-like (see the exact definition in Section 2). In this sense classical null curves are 1-degenerate curves. We obtain a reference along an s-degenerate curve in an n-dimensional Lorentzian space with the minimum number of curvatures. That reference generalizes the reference of Bonnor for null curves in Minkowski space–time and it will be called the Cartan frame of the curve. The associated curvature functions are called the Cartan curvatures of the curve. We characterize the s-degenerate helices (i.e. s-degenerate curves with constant Cartan curvatures) in n-dimensional Lorentzian space forms and we obtain a complete classification of them in dimension four.

A criterion for reduction of variables in the Willmore-Chen variational problem and its applications

We exhibit a criterion for a reduction of variables for WillmoreChen submanifolds in conformal classes associated with generalized KaluzaKlein metrics on flat principal fibre bundles. Our method relates the variational problem of Willmore-Chen with an elasticity functional defined for closed curves in the base space. The main ideas involve the extrinsic conformal invariance of the Willmore-Chen functional, the large symmetry group of generalized Kaluza-Klein metrics and the principle of symmetric criticality. We also obtain interesting families of elasticae in both lens spaces and surfaces of revolution (Riemannian and Lorentzian). We give applications to the construction of explicit examples of isolated Willmore-Chen submanifolds, discrete families of Willmore-Chen submanifolds and foliations whose leaves are Willmore-Chen submanifolds.

Solutions of the Betchov-Da Rios soliton equation: a Lorentzian approach

Abstract The purpose of this paper is to find out explicit solutions of the Betchov-Da Rios soliton equation in three-dimensional Lorentzian space forms. We start with non-null curves and obtain solutions living in certain flat ruled surfaces in l3 and h13, as well as in r3 and s3. Next we take a null curve and have got solutions lying in the associated B-scrolls in l3, s13 and h13. It should be pointed out that we extend previous results already obtained, and as far as we know, this is the first time that solutions in the De Sitter 3-space appear in the literature. Soliton solutions are characterized as null geodesics in B-scrolls.

HYPERSURFACES IN THE NON-FLAT LORENTZIAN SPACE FORMS WITH A CHARACTERISTIC EIGENVECTOR FIELD

In a series of early papers, with the aim of knowing of the shape of a pseudo-Riemannian hypersurface satisfying a certain differential equation in the induced Laplacian, we found a remarkable family of hypersurfaces in the Lorentz-Minkowski space whose mean curvature vector is an eigenvector of the Laplacian. Actually, the last two authors showed in [8] that the equation ∆H = λH , for a real constant λ, characterizes the family of surfaces in L3 made up of the quite interesting B-scrolls and the so-called standard examples, as well as minimal surfaces. Looking at those results obtained for surfaces in L3, the following geometric question was stated in [9] for hypersurfaces in Ln+1 (n > 2): Does the equation ∆H = λH mean that both the mean and the scalar curvatures of the hypersurface are constant? We were able to give a partial solution to that problem, since we had needed to do an additional hypothesis on the degree of the minimal polynomial of the shape operator. It is worth pointing out that the additional assumption was mainly made to control the position vector field of the hypersurface into Ln+1. Now when the ambient space is a non-flat pseudoRiemannian space form, Sn+1 ν (r) or Hn+1 ν (r), then the hypersurface is of codimension two in Rn+2 ν or R ν+1 , respectively, but Sn+1 ν (r) and Hn+1 ν (r) being both totally umbilical hypersurfaces in the corresponding pseudo-Euclidean space, it seems reasonable to hope for a richer classification of hypersurfaces into those spaces by means of the equation ∆H = λH . Or even, one looks for getting a complete answer to the stated problem in non-flat ambient spaces. In this paper we give a classification of surfaces in the 3-dimensional non-flat Lorentzian space forms satisfying the equation ∆H = λH . We show that the family of such surfaces consists of minimal, totally umbilical and B-scroll surfaces. As for hypersurfaces we suppose that their shape operators have no complex eigenvalues. This condition does not seem as restrictive as one could think, in view of examples and results given in section 5. Actually, we find that family is set up by minimal, totally umbilical and so-called generalized umbilical hypersurfaces, which are nothing but a natural generalization of B-scrolls.