Null Curve Evolution in Four-Dimensional Pseudo-Euclidean Spaces

del Amor, José; Giménez, Ángel; Lucas, Pascual

doi:https://doi.org/10.1155/2016/5725234

Advances in Mathematical Physics

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Research Article | Open Access

Volume 2016 | Article ID 5725234 | https://doi.org/10.1155/2016/5725234

Null Curve Evolution in Four-Dimensional Pseudo-Euclidean Spaces

José del Amor,¹Ángel Giménez,²and Pascual Lucas¹

Academic Editor: Emilio Elizalde

Received13 Apr 2016

Accepted05 May 2016

Published02 Jun 2016

Abstract

We define a Lie bracket on a certain set of local vector fields along a null curve in a 4-dimensional semi-Riemannian space form. This Lie bracket will be employed to study integrability properties of evolution equations for null curves in a pseudo-Euclidean space. In particular, a geometric recursion operator generating infinitely many local symmetries for the null localized induction equation is provided.

1. Introduction

Recently in [1, 2] a connection between the local motion of a null curve in and the celebrated KdV equation was given. In [3] the author obtained a connection between a null curve evolution in (to which we refer as the “null localized induction equation” or NLIE), and the Hirota-Satsuma coupled KdV (HS-cKdV) system, we briefly remind the reader of here. Hirota and Satsuma [4] proposed (perhaps up to rescaling) the HS-cKdV systemwhich describes the interactions of two long waves with different dispersion relations. Many systematic methods have been employed in the literature to clarify the integrability of the HS-cKdV system: the Lax pair [5–7], the Bäcklund transformation method [8], the Darboux transformation [9–11], the Painlevé analysis [6, 7], the search of infinitely many symmetries and conservation laws [4, 12, 13], and so forth. Fuchssteiner [12] discovered that HS-cKdV system given by (1) admits the symplectic and the cosymplectic operators:Furthermore, an infinite hierarchy of symmetries for (1) was found in [13]:In this case the recursion operator is not hereditary. Hence, the bi-Hamiltonian formulation of the HS-cKdV system does not arise from a Hamiltonian pair.

In this paper, we extend some of the results given in [1–3] to a more general background. More specifically, we generalize the Lie algebra structure defined on the local vector fields along null curves from the 3-dimensional Minkowski space to 4-dimensional semi-Riemannian space forms of index or and curvature . This Lie algebra together with the properties about the HS-cKdV system described above will be used to construct an infinity hierarchy of commuting symmetries for the NLIE equation in a 4-dimensional pseudo-Euclidean space.

It is interesting to point out that, from a physical point of view, the 4-dimensional space is a more realistic context than the -dimensional background, the latter very often serving merely as a toy model. Let us recall that relativistic particles models have been described by actions defined on null curves whose Lagrangians are functions of their curvatures [14, 15]. These actions were also studied in Minkowski spaces and (see [16, 17]), as well as in -dimensional Lorentzian space forms in [18]. All these works were addressed to study variational problems on null curve spaces, and they have shown that the underlying mechanical system is governed by a stationary system of Korteweg-de Vries type. Furthermore, if is a critical point (the so-called null elastica) for the action , where is a constant, then the associated solution to the NLIE starting from is the null elastica evolving by rigid motions in the direction , where is actually the rotational Killing vector field for the null elastica (see [16, 18]), and it served to determine the benchmark for the evolution equation NLIE in [1]. This idea was originally explored by Hasimoto between the elastica (an equilibrium shape of an elastic rod) and the “localized induction equation” (LIE). As might be expected, relationships with the Korteweg-de Vries evolution systems still arise when the null curve motions in 4-dimensional backgrounds are considered.

One of the many advantages of having a scalar evolution equation coming from a curve motion is that many aspects of its integrability can be elucidated from the intrinsic geometry of the involved curves (see [19, 20]). Conversely, integrability properties of the curvature flow can be employed to determine integrability properties of the curve evolution equation (see [1, 21–23]). Despite the numerous well-known connections between curve evolution equations and integrable Hamiltonian systems of PDEs, there is still a lack of understanding regarding the mechanisms and links among the different frameworks. Our overall aim here is to go further into those concerns.

The rest of this paper is organized as follows. In Section 2 we summarize some basic notions about formal variational calculus on which the Hamiltonian theory of nonlinear evolution equations is based. In Section 3 we have included some background formulas and results concerning the differential geometry of null curves in a semi-Riemannian space form. In particular, we study the properties of variation vector fields along a null curve in a semi-Riemannian space form as well as the variational formulas for its curvatures. In Section 4, the Lie bracket on the set of -local vector fields locally preserving the causal character along null curves in a 3-dimensional Lorentzian space given in [1] is extended to 4-dimensional semi-Riemannian space forms. This section also includes a discussion about the connection between the geometric variational formulas for curvatures and the Hamiltonian structure for the HS-cKdV system. The above results will be employed in Section 5 to introduce the NLIE equation as a geometric realization of HS-cKdV equations and to construct a geometric recursion operator generating an infinity hierarchy of commuting symmetries for the NLIE equation.

2. Preliminaries

In this section we summarize some necessary notions and basic definitions from differential calculus which are relevant to the rest of the paper (see [24–26] for a very complete treatment of the subject). Let be a positive integer and consider differentiable functions in the real variable . Set

Let be the algebra of polynomials in and their derivatives of arbitrary order, namely, We refer to the elements of whose constant term vanishes as . Acting on the algebra is defined as derivation obeying: thereby becoming a differential algebra.

Remark 1. In a more general setting we can consider , for example, to be the algebra of local functions, that is, , where is the algebra of locally analytic functions of and their derivatives up to order (see [27–30]). All the results and formulas established in Sections 4 and 5 involving the algebra remain valid if the differential algebra of polynomials is replaced by the differential algebra of local functions. Nonetheless, the differential algebra of polynomials is sufficient for our purposes.

It is customary to take as the total derivative which can be viewed as In addition to , other derivations may also be considered. The action of is determined if we know how acts on the generators of the algebra. Indeed, set then, for any we have The space of all derivations on , denoted by , is a Lie algebra with respect to the usual commutator Derivations commuting with the total derivative have important properties. Among others, if , we have Thus where . Let be an element of and writeThe set of derivations is a Lie subalgebra of , and it induces a Lie algebra on the space . Indeed, a direct computation shows that are derivations in verifying , where ; that is, This latter commutator can also be expressed with the aid of Fréchet derivatives as , whereWe will refer to as an evolution derivation (or a vector field, provided that no confusion is possible), and the algebra of all evolution derivations will be denoted by . Observe that, in particular, if we take , then , with .

Set for . Consider an evolution equation of the formwhere is an element of . An element is called a symmetry of the evolution equation (16) if and only if . Symmetries of integrable equations can often be generated by recursion operators which are linear operators mapping a symmetry to a new symmetry. A linear differential operator is a recursion operator for the evolution equation (16) if it is invariant under , that is, , where is the Lie derivative acting as for all . is said to be hereditary if for an arbitrary vector field the relation is verified.

3. Null Curve Variations in

The geometry of null curves is quite different from the nonnull ones, so let us review the relevant results, going further into what concerns us most for later work.

A semi-Riemannian manifold is an -dimensional differentiable manifold endowed with a nondegenerate metric tensor with signature . The metric tensor will be also denoted by and the Levi-Civita connection by . The sectional curvature of a nondegenerate plane generated by is where is the semi-Riemannian curvature tensor given by

Semi-Riemannian manifolds with constant sectional curvature are called semi-Riemannian space forms. It is a well-known fact that the curvature tensor adopts a simple formula in these manifolds:where is the constant sectional curvature. When the curvature vanishes, then is called pseudo-Euclidean space and will be denoted by .

Let denote a 4-dimensional semi-Riemannian space form with index , background gravitational field , and Levi-Civita connection . A tangent vector is time-like if , space-like if or , and null if . Therefore, a parametrized curve is called null if its tangent vector is null at all points in the curve. Fixing a constant , we can consider (if is not a geodesic) the parameter given by where is any parameter. When this parameter agrees with the pseudo arc-length parameter for the null curve. In fact, it is easy to show that is nothing but a linear reparametrization of the pseudo arc-length parameter and it verifies

Throughout this paper it will be supposed that we have fixed a constant , will be denoted by , and we will also refer to it as the pseudo arc-length parameter. The Cartan frame of a nongeodesic null curve , verifying that is linearly independent for all , is given by , where with . The Cartan equations readwhere denotes the covariant derivative along and , are the curvatures of the curve. The fundamental theorem for null curves tells us that and determine completely the null curve up to semi-Riemannian isometries (see [31]). Even more, if functions and are given we can always construct a null curve, pseudo arc-length parametrized, whose curvature functions are precisely and . Then any local scalar geometrical invariant defined along a null curve can always be expressed as a function of its curvatures and derivatives. A nongeodesic null curve being pseudo arc-length parametrized and admitting a Cartan frame as above is called a Cartan curve. The bundle given by is known as the screen bundle of (see [31]). Projections of the variation vector fields onto the screen bundle will play a leading role in this research.

Let be a null curve, for the sake of simplicity the letter will also denote a variation of null curves (null variation) with the initial null curve. Associated with such a variation is the variation vector field , where . We denote by the differentiable function verifying and by the covariant derivative along the curves . We write , , , and so forth, for the corresponding objects in the pseudo arc-length parametrization.

Definition 2. Let be the set of smooth vector fields along . We say that locally preserves the causal character if . We also say that locally preserves the pseudo arc-length parameter along if satisfies .

The following properties for null variations can be found in [17] when , but they can be easily adapted to the general situation.

Lemma 3. If is a null variation, then its variation vector field verifieswhere .

Thus we obtain that locally preserves the causal character and, moreover, locally preserves the pseudo arc-length parameter if and only if , which in such a case also entails commutation of and . We define some functions that will play a key role in the rest of the paper, namely, given a vector field we consider the following projections of and on the screen bundle given by

Lemma 4. With the above notation, the following assertions hold:(a);(b),(c),(d),(e) − ,(f), where

Proof. Set the covariant derivative. From (24) we obtainwhere . Now, taking into account formulas (18), (19), and (27) we have Considering again (18), (19), (27), and (28) we deduceSince , the tangent component of vanishes and the expression for becomesAs a consequence the vector field boils down toFinally, a similar computation leads toIn the same way, since the component of in vanishes, we deduce

Consider the space of pseudo arc-length parametrized null curves in . For , it is easy to see that is the set of all vector fields associated with variations of pseudo arc-length parametrized null curves starting from . It is clear that a vector field in locally preserves the causal character and the pseudo arc-length parameter. The converse can also be proved applying a similar procedure as in [2].

Proposition 5. A vector field along is tangent to if and only if it locally preserves the causal character and the pseudo arc-length parameter; that is,Consequently, if is expressed by , where , , , and are smooth functions, then if and only ifwhere is a formal indefinite -integral. Furthermore,

Proof. For a generic vector field we obtainIf , Lemma 3 implies thatIn such a case, by using (37) and (38), we deduce that Last equations easily give rise to (35). Expression (35) becomes and the following holds:Replacing into (42) and rearranging terms, we easily obtain (36). Conversely, if is a vector field verifying (39), then it arises from an infinitesimal variation of null curves. Indeed, according to the Cartan equations (23), Lemma 4, and formulas (35) and (42), we consider the matrices verifyingwhere where the functions and are given by formulas (e) and (f) given in Lemma 4, respectively, with . Following the same procedure as described in Lemma 1 of [2] we can construct a null curve variation of whose variation vector field is .

From Proposition 5, a tangent vector field and its covariant derivative are expressed by

Remark 6. Observe that a tangent vector field is completely determined by the differentiable functions and and two constants, since the operator is used twice; once for obtaining from and once more for obtaining from and . Therefore, given two differentiable functions and and two constants, we can construct a vector field locally preserving pseudo arc-length parameter along whose projections on the screen bundle are precisely and . Both constants could be determined or related if constraints on null curve variation are considered, but, for our algebraic purposes, we will consider generic constants.

4. A Lie Algebra Structure on Local Vector Fields

Our objective now is to define a Lie algebra structure on the set of local vector fields which locally preserve the causal character. To this end, we need first to set up the spaces in which we are going to work. Let and be smooth functions defined on an interval and set the real algebra of polynomials in , and their derivatives of arbitrary order; that is, where and .

Let be a null curve with curvatures and and consider the set of vector fields along whose components are polynomial functions An element of will be called a -local vector field along . The set of -local vector fields (locally preserving the causal character) will be denoted by and within it, the -local variation vector fields locally preserving pseudo arc-length parameter are described as In this context, from Proposition 5 and taking into account Remark 6, we can explicitly calculate the -local pseudo arc-length preserving variation vector fields by means of its constants of integration.

Proposition 7. Let be a vector field in , then if and only if it is fulfilled that where , are constants and is the antiderivative operator verifying that when acting on .

Consequently, let us consider the set Given a pair of functions and two constants and , we will denote by the -local pseudo arc-length preserving variation vector field

Example 8. Consider the pair of functions ; then . In particular, if we take and , where and are constants, one obtains the vector fields The vector fields and will be the starting point of the commuting hierarchy of symmetries in Section 5.

Note that to introduce the concept of symmetry (and so a recursion operator) and furnish the phase space of null curve motions with a formal variational calculus in Section 5, an appropriate Lie bracket on the set of local vector fields should be defined. To this end, we first introduce a convenient derivation on both the differential algebra and the local vector fields along a null curve. Motivated by [32] and bearing in mind Lemma 4, given , we denote by the unique tensor derivation fulfilling:where , , and are given in (26). We now restrict our definition of Lie bracket only on the set , which will be enough for our purposes.

Proposition 9. Let be a null curve in and consider the map given by Then the following holds:(a) is well defined; that is, if then .(b) for all .(c) is skew-symmetric.(d)For and we have (e) satisfies the Jacobi identity.(f) is closed for elements in , that is, if , then .

Proof. Given two vector fields and in , set . The components and of are given by Since , they verify formulas (35) and (42) which, together with definitions of and , lead to the relation . The latter is the condition equivalent to , thus proving (a). To prove (b), it is sufficient to check the same equality solely for generators and of algebra . We will calculate the expressions of , , and (corresponding functions to the Lie bracket ), by means of , , and (corresponding functions to vector fields ). If is any vector field, we haveBearing in mind relations (61) and expressions of and obtained in Lemma 4, we deduceBecause of symmetry of formulas (62), deleting terms with repeated factors and rearranging the other, we obtainFrom Lemma 4 we obtainExpanding each earlier term by using and other properties it follows thatBased on the expressions and it is also easy to establish thatWhen adding up (64), (65), and (66) and making a long but easy computation we obtain Using again Lemma 4 we haveIn the same way as (63), we can compute the terms of (68),After some work, it also follows from (69) that Paragraph (c) is a direct consequence of the definition and (f) is also trivial taking into account the expression for given in (63). Paragraph (d) follows from a straightforward computation. Finally, to prove (e), let us denote . By using (d) we obtain

In particular, Proposition 9 entails that the set of -local pseudo arc-length preserving variation vector fields is a Lie subalgebra of the Lie algebra of the -local vector fields .

Before turning to study the geometric hierarchies of null curve flows in Section 5, we point out that (36) and (56) are particularly noteworthy when vector fields locally preserve the pseudo arc-length parameter and curvature vanishes. Equations (36) and (56) may be rewritten aswherewith , , and . It should be remarked that and come very close to being the symplectic and cosymplectic operators, respectively, for (up to scaling) the Hirota-Satsuma system (see [12, 13]). Equation (72) can also be regarded aswhere

It is now therefore evident that and are the symplectic and cosymplectic operators, respectively, for a rescaling of the HS-cKdV system. They have been obtained in a natural way using projections onto the screen bundle of both, the variation vector field and its covariant derivative . This allows us automatically to determine the recursion operator , and the crucial relation

Somewhat analogous relationships were obtained between curve evolution in 3-dimensional Riemannian manifolds in [19] (or more generally in -dimensional Riemannian manifold with constant curvature in [20]) and the mKdV system. The above connection together with the availability of the Lie bracket provided by Proposition 9 will be employed to study the integrability of null curve evolution in the next section.

5. Geometric Hierarchies of Null Curve Flows

The background given in [1] for the 3-dimensional case used to construct a commuting hierarchy for null curve evolutions can also be well adapted to the 4-dimensional case. Consider the space of pseudo arc-length parametrized null curves in the pseudo-Euclidean space . A map is referred to a scalar field on and will be also denoted by . Let be the algebra of -valued scalar fields on , that is, if , then for all . In this sense, we will also understand the curvatures scalar fields with its obvious meaning.

Similarly, a map is referred to as a vector field on , and will be also denoted by . We will denote the set of tangent vector fields on as , and within we consider the subset of vector fields such that ; namely, if we denote , thenwhere the derivative and antiderivative operators act on scalar fields as and , respectively. Thus stands for the set of -local vector fields locally preserving the pseudo arc-length parameter and the causal character. These vector fields commute with the tangent vector field , so they will be called evolution vector fields. We also denote by and the sets of vector fields such that and , respectively. Hence,

Remark 10. In what follows, we will operate with scalar fields and vector fields in the natural way, understanding that the result of the operation is again a scalar field or vector field. For instance, if are vector fields on , then is a scalar field, where ; or is again a vector field, where and so on.

Hence, for , the operator is defined as . The operator can also be described in other words when . Consider is a null curve in and suppose that , then In fact, the tensor derivation is an extension of the Fréchet derivative defined in (15) for derivations to vector fields on the null curves space. In this way, this operator can be easily translated to the context of any other type of curves.

The Lie algebra structure on local vector fields locally preserving the causal character provided by Proposition 9 (along a particular curve) can also be easily extended on the set .

Proposition 11. The map given by is a Lie bracket verifying the following:(a) for all .(b) is closed for elements in ; that is, if , then . Hence, is a Lie bracket, is a Lie algebra, and the space of evolution vector fields is a Lie subalgebra of .

Let us define as the set of derivations on defined in the natural way and the Lie subalgebra of all evolution derivations. In this setting, the elements of are given by , with , such that they are defined as usual by , for all and . Each vector field on can be regarded as a derivation on , acting on the generators and in the following way:

Theorem 12. The map defined by where , , and is defined in (75), is a one-to-one homomorphism of Lie algebras. In particular, and are commuting vector fields with respect to the Lie bracket defined by Proposition 11 if and only if their corresponding curvature flows and commute with respect to the usual Lie bracket for scalar fields.

Proof. Since the map is clearly linear, it is enough to prove that keeps the Lie bracket, that is, , the latter being equivalent to show thatFrom the last equality of Proposition 9(b) we deduce and it is therefore satisfied for all . Finally, formula (83) is followed from In order to prove the injectivity we will prove that implies , which is equivalent to proving that implies . As a first step we will prove that if , then and so, from formula (46), . According to (74) we have that For a scalar field we denote by the order of the highest derivative (with respect to both or ) appearing in ; that is, Suppose that ; then we have that . Since it is necessarily obtained that , whence . Accordingly, , which together with the equation would lead to and so a contradiction. Therefore the scalar field is constant, , and it would verify the equation The latter equation is satisfied if and only if and , that is, . Using formula (37), the equation can be developed asSuppose that ; then (90) give rise to the following implications: Nevertheless, those orders of derivation represent a direct contradiction to the first equation in (90) unless .

Remark 13. From Theorem 12 we have that is a Lie subalgebra of the algebra of all evolution derivations. Thus, we conclude that the algebra of evolution vector fields on can be regarded as a Lie subalgebra of the evolution derivations.

Consider the vector fields and borrowed from Example 8. Their flows are governed by which in turn induce evolutions for the curvature functions and given byObserve that the evolution equation (95) is the Hirota-Satsuma equation (1) (up to scaling the variables , and rescaling the functions , ). Besides, the flows associated to the curvatures given by and are basically flows and given in (3). We refer to (93) induced by (which also appears in [3]) as the null localized induction equation (NLIE). Theorem 12 will be used below to obtain a recursion operator for NLIE and thereby prove its integrability.

Proposition 14. The operator acting on symmetries ,is a recursion operator for NLIE.

Proof. Let . Then by definition and making use of (76) we obtain The result can be easily deduced as a consequence of Theorem 12.

We now proceed with the construction of an infinite hierarchy of symmetries following the same scheme as in (3):Then, we have and the corresponding curvature flow isLikewise, the next vector field in the hierarchy becomes

Note that the above geometric hierarchy of commuting vector fields at the curve level is a generalization of the ones obtained in [1] for the 3-dimensional case, albeit using a different procedure. In fact, it was not possible to extend the procedure used in [1] to obtain the recursion operator and the Hamiltonian structure at the curve level to the 4-dimensional setting, mainly because of the appearance of nonlocal vector fields. Searching for a Hamiltonian structure at the curve level for the 4-dimensional case will be one of the subject for future research.

6. Conclusions

In this paper, our primary aim was to study the integrability properties of null curve evolutions in a flat 4-dimensional background. We undertook our research in an enough degree of generality for the purpose of showing the role of the constants appearing on it, especially when they possess geometrical meaning. In that regard, the way in which the computations were conducted to expose the most important elements of the Hamiltonian structure for curvature flows is particularly important. One of the most surprising facts was to obtain the recursion operator (split into both the Poisson operator and the symplectic operator in formulas (74) and (75)) of the Hirota-Satsuma system by means of the geometry of null curves or, more precisely, making use of the projection of convenient variation vector fields onto the screen bundle. Similar results were obtain in [19, 20], suggesting that the screen bundle of a null curve may be thought of as playing the same role of the normal bundle in a Riemannian curve. We can therefore also state the following important conclusion: if we have a evolution vector field with having the property of being the gradient of a certain functional , then the flow associated to the curvatures is a completely integrable Hamiltonian system.

Furthermore, in Proposition 14 we have lifted the recursion operator for the Hirota-Satsuma system (at the curvature level) to a recursion operator for the NLIE equation (at the curve level), enabling us to obtain an infinite hierarchy of commuting vector fields. Proposition 11 shows that the subspace consisting of -local evolution vector fields (denoted by ) is closed under bracket and contains the commuting flows as a subalgebra.

One of the many benefits of increasing the dimension of the ambient space has been that the connections between integrable hierarchies of both null curves and their curvature flows become clearer. Nevertheless, finding a Hamiltonian structure at the curve level still needs to be achieved. In addition, it would be interesting to develop a purely geometric method to construct the existing structures of the dynamic of null curve motions without lifting any element from the curvature flow. Accordingly, further work is needed, perhaps in a nonlocal background, if possible, to properly understand which also has appeared in different contexts.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work has been partially supported by MINECO-FEDER (Ministerio de Economía y Competitividad-FEDER) Project MTM2015-65430-P and Fundación Séneca (Región de Murcia) Project 19901/GERM/15, Spain.

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Copyright © 2016 José del Amor et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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