Binormal Motion of Curves with Constant Torsion in 3-Spaces

Abstract

We study curve motion by the binormal flow with curvature and torsion depending velocity and sweeping out immersed surfaces. Using the Gauss-Codazzi equations, we obtain filaments evolving with constant torsion which arise from extremal curves of curvature energy functionals. They are “soliton” solutions in the sense that they evolve without changing shape.

1. Introduction

A large class of physical systems are modelled in terms of motion of curves and surfaces in Euclidean space . A remarkable example is the so-called localized induction equation (LIE)which is a soliton equation used to model the dynamics of a thin vortex filament in an incompressible, inviscid, homogeneous, -dimensional fluid [1–3]. Quite often, by resorting to the underlying geometry one can gain considerable insight into the dynamics of physical systems [3, 4]. Here, we use a geometrical approach to investigate an extension of (1) obtained by considering a smooth map , , verifyingwhere is a suitable smooth function, denotes the Levi-Civita connection on , , and is a Riemannian or Lorentzian () -space form with constant curvature ; that is, is one of the following: , the sphere , the hyperbolic space , the Minkowski space , the de Sitter space , or the anti de Sitter space .

Under mild conditions we will see that a curve motion following (2) describes a curve evolving under the binormal flow, with velocity depending on curvature and torsion (19), and determines an immersed surface, , in . Then, fundamental results of the theory of submanifolds can be applied and it will be seen that solving geometrically (2) amounts to solving the Gauss-Codazzi equations (40) and (41), since that would give us the curvature and torsion of a geodesic foliation of . Alternatively, one can determine the evolution by finding a single solution, working as initial condition , and then giving a geometrical description of the binormal flow.

If and , (2) reduces to LIE (1) and it can be seen that Gauss-Codazzi equations boil down to Da Rios equations found in 1906 [2]. In Lorentzian backgrounds (1) has been studied in [1, 5], while long time existence of closed solutions in Riemannian ambient spaces are analyzed in [6]. If , travelling wave solutions of the Gauss-Codazzi equations have been investigated in [7]. In the second part of this work, we focus on curves evolving by (2) with constant torsion and use the Gauss-Codazzi equations to construct solutions by means of extremal curves for curvature dependent energies and associated -parameter groups of isometries.

2. Preliminaries

Consider the Euclidean semispace , that is, endowed with the canonical metric of index , denoted by , and the Levi-Civita connection, denoted by . Then, the semi-Riemannian -space forms (the Riemannian case, will be simply denoted by ) can be isometrically immersed in , the -dimensional Euclidean semispace, in a standard way [8]. The flat case, ,  ,  , corresponds to either or the Minkowski space . They can be isometrically immersed in endorsed with the metricin an obvious manner:

When , correspond to the -sphere,   , and the de Sitter -space,   , defined bywhere . Finally, for we obtain the hyperbolic -space, , and the anti de Sitter -space, ()

The standard isometric immersions of into ([8] p. 20) will be all denoted by and the induced metrics also by , while the Levi-Civita connections on and are denoted by and , respectively. As usual, the cross product of two vector fields in , denoted by   ×  , is defined so that for any other vector field of , where stands for the determinant.

Now, for a given isometric immersion of a surface, , , we denote by the Levi-Civita connection of the immersion . As it is also customary, for a surface in any -dimensional space form , we require the first fundamental form to be nondegenerate. Take tangent vector fields to and choose a normal vector field to in . Then the formulas of Gauss and Weingarten are, respectively, as follows [8]:where is the position vector, denotes the second fundamental form of in , stands for the Weingarten map, and denotes the connection on the normal bundle of . By using (7) and (8) and denoting by and the Riemann curvature tensors associated with and , respectively, the following relation holds:

while the equations of Gauss and Codazzi are given, respectively, by [8]where is defined by

Now choosing an adapted local orthonormal frame in such that the vectors , are tangent to and is normal to in , and denoting by the dual frame of , the Cartan connection forms are defined byfor , where is the causal character of . Then, and

[8]. We will often resort to the standard abuse of notation and identification tricks in submanifold theory.

3. Binormal Evolution Surfaces

The covariant derivative of a vector field along a curve will be denoted by . Let be a unit speed nongeodesic curve immersed in with nonnull velocity , ; therefore, it is assumed to be either spacelike or timelike. If it also has nonnull acceleration , then is a Frenet curve of rank or and the standard Frenet frame along is given by , where is chosen so that . Then the Frenet equations define the curvature, , and torsion, , along where , , is the causal character of , , and , respectively. Notice that the following relations hold:

Curves for which both curvature and torsion are constant are called Frenet helices. In a semi-Riemannian space form any local geometrical scalar defined along Frenet curves can always be expressed as a function of their curvatures and derivatives.

Given a smooth map , , satisfying (2), we usually identify and . Assume that the initial condition is a unit speed Frenet curve of rank or ; then , which will be called the filament at time  , is also unit speed parametrized . In fact, we have , where the last equality is obtained from (2). So, since , then so is ; that is, (2) is a length-preserving evolution. Assuming also that is nonnull everywhere, the associated Frenet frame will be defined for all and combining (2) and (15) we obtainThis means that evolves by the binormal flow with velocity  . We are going to suppose also that is never zero so that defines an immersed surface in swept out by . It will be denoted by and called a binormal evolution surface (BES) with initial condition    and velocity  . The curves , perpendicular to the filaments, are called fibers of . The time variation of the Frenet frames is described in the following proposition (cf. of [3] for surfaces in ).

Proposition 1. Let be a BES of with velocity (19). Thenwhere and and denote the curvature and torsion of the curves .

Proof. Under our assumptions, all are unit speed parametrized and they all have well defined Frenet frame satisfying (15)–(17), so (20) is clear. As for the second part, since is not a geodesic in and is not null, then, for sufficiently small , the unit Frenet normal to , , is parallel to a (local) unit normal to . This means that are geodesics in for any and the parametrizationdetermines a geodesic coordinate system with respect to which the metric can be written as Now, the Gauss and Weingarten formulas (7) and (8), in combination with the Gauss and Codazzi equations (10) and (11), will give us all the relevant geometric information about the immersion . This requires bringing in some computational stuff and very long calculations whose details are omitted here. Thus, the Christoffel symbols of the Levi-Civita connection of (23) with respect to the parametrization (22) (see, e.g., [8], Proposition ) can be computed from the metric coefficients . In our case, we havewhere subscripts and mean partial derivative with respect to and , respectively. This makes it possible to know the expression for the Levi-Civita connection of ([8], 1.4), denoted here by As before, represent the Frenet frames along , and we choose the following local adapted frame on :where is the unit normal to (locally defined). Then, combining (7), (8), (13), (14), and (15)–(17), one gets where and denote the curvature and torsion of the curves .
The second fundamental form can be considered as a quadratic form given by ; therefore, we obtain from (27) thatwith respect to the parametrization (22), where is the coefficient of the second fundamental form ([8], 2.3) of in given by . Since is determined by , can be computed with the aid of (25) and the Gauss formula (7) givingUsing again the Gauss and Weingarten formulas (7) and (8), it can be shown that satisfies the following PDE system:This system can be expressed equivalently in terms of the time variation of the Frenet frame. In fact, from the Gauss formula (10) and (30), we haveSo, differentiating and using once more Gauss and Weingarten formulas, we have which combined with (32) givesFinally, from (33), (35), and the cross product relations (18), one has which ends the proof.

Now, combining (9) and Gauss and Codazzi equations (10) and (11) with (13), (25), (26), and (27), we obtain after a long computationNotice that the Gauss equation (37) is equivalent to (29). By substitution of (29) in (39) we get that the Codazzi equations (38) and (39) for boil down to

Observe that if , (40) and (41) are precisely Da Rios equations for the vortex filament [2]. Moreover, (40) and (41) are the compatibility conditions of the PDE system (30)–(32). Thus, from the fundamental theorem of submanifolds ([8], 2.7), given functions , , and , smoothly defined on a connected domain and satisfying (40) and (41), there exists a solution of (30)–(32) (and, consequently, of (20)-(21)) determining a smooth isometric immersion (unique up to rigid motions, if is simply connected) of a surface in whose metric and the second fundamental form are given, respectively, by and , where is obtained from (29).

Computing the Christoffel symbols from the metric coefficients for such an immersion, , we see that are arc-length parametrized geodesics . Then, a combination of the Gauss formula (7) and (15)–(17) for the Frenet frame along the coordinate curves , , shows that the unit Frenet normals are perpendicular to the surface . Hence, , but then the second coefficient of (23) implies that and is a solution of (19). Since is foliated by geodesics having and as curvature and torsion, respectively, the immersion itself, , is geometrically determined by and , because, from the fundamental theorem of curves, for any fixed , there exists a unique curve (up to congruences and causal character of the Frenet frame) having and as curvature and torsion. Then, smooth assembling of these curves ,  , would give . So, geometrically solving (19) amounts to solving system (40), (41).

Another consequence of the Gauss-Codazzi equations (40) and (41) is that, besides length, other geometric quantities may also be invariant for closed filaments. More precisely, we have the following.

Proposition 2. Let be a binormal evolution surface in with velocity and assume that and all filament curves are -closed in . If , then and , with , , are independent of . Furthermore, if is constant, then also is invariant. Finally, if , then does not depend on .

Proof. Only first part is proved since the others are similar. If , then, the invariance is a direct consequence of , because using (41) we have . Now, from (40)

4. Evolution with Constant Torsion

Now we study binormal evolution surfaces, whose filaments have the same constant torsion. Since , . Choose so that . Assume first .

Proposition 3. Let be a binormal evolution surface, whose filaments satisfy . Then is extremal for the energy and , where is a -parameter group of isometries of . Moreover, the fibers of have constant curvature and zero torsion (if they are not geodesics) in . In particular, if , are either ruled surfaces or rotational surfaces.

Proof. By substituting in (40) we have and the metric with respect to the chosen coordinate system is . This means that is a warped product surface [8], and since , we have that is a Killing field of . Now, integrating (41) we getfor some . Moreover, since , we have that (43) is the Euler-Lagrange equation for ([9, 10]) and must be an extremal of in , . On the other hand, for a given field along , , the following variation formulas for , , and can be obtained using standard computations and the Frenet equations (see, [9, 10]) where the subscript denotes differentiation with respect to the arc-length. Now, combining (43), (44), and the Frenet equations, one can see thatalong , where . This means that is a Killing field along ([9–11]), but this field is precisely . Now, (44) imply that the Killing vector fields along a curve form a six-dimensional linear space. Moreover, the Lie algebra of is six-dimensional and the restriction of a Killing vector field in to any curve gives a Killing vector field along . Hence, every Killing vector field along a curve, , is the restriction to of a Killing vector field of [11]; in other words, can be extended to a Killing field on (denoted also by ). Hence, the associated -parameter group is formed by isometries of and is obtained as , where . Since , , and , we get that does not depend on . Moreover, as fibers are orbits of a Killing field of , they have constant curvature. Now, for any take an arc-length parametrization, , of the fiber of through . With the subscript denoting the geometric elements associated with the curve , we have and, using the last equation of (21), we obtain if has nonnull acceleration. Thus, differentiating (46) with respect to and using again (21), we have that must verifyfrom which we see that either and is a geodesic in or and is a planar circle.
On the other hand, if has null acceleration and is not a geodesic, then we can consider the following frame along . Define as the lightlike field on given by and denote by the only lightlike vector such that and . In this case, we have the following equations:for certain function which will be also called the torsion of (here, the “curvature” is considered to be ). Then, from the second equation of (21), it is clear that .
Finally, we restrict ourselves to flat ambient spaces or . For simplicity, we take first . If , , then any planar curve is critical for [12] and must be a right cylinder shaped on . Assume then that , . Then the Killing field along , , can be written as , for some , , and a constant vector in . By scalar multiplication of with the covariant derivative of along , we obtain and, then, is a rotational surface in with profile curve . These facts can be extended to the Minkowski space by similar arguments.

Moreover, if , , it is not difficult to see that, identifying the plane containing and , choosing a coordinate system containing , and using , it is possible to find a coordinate system in where and , for some constant . This also works in the Minkowski space . In particular, take an extremal curve for either in or in and choose a coordinate system where and , for some constant . Assuming , thenwhere verifies that are rotation surfaces with planar filaments evolving by (19). If the extremal curve lies in the Minkowski plane , then choosing the same coordinate system as before, one could also construct a surface with planar filaments evolving by (19) by rotating around a lightlike axis. In fact, suppose without loss of generality that the lightlike axis is determined by ; then, the parametrization of is given by

where verifies . In this case, fibers are spacelike curves with null acceleration.

We remark that a periodic solution of (43) does not determine a closed filament necessarily, and closure conditions have to be derived. For instance, if is a periodic solution of (19) with period and is the corresponding curve in , then integrating (43) we have , for some . Choosing a coordinate system in where and , we see that is also periodic. Using the fact that is arc-length-parametrized, we have . Hence, closes up, if and only if .

Take now . If is also constant, filaments are Frenet helices. The unit binormal is a Killing vector field on Frenet helices; hence, their evolution under the -flow satisfies , so we assume is not constant. Then (40) suggests studying travelling wave solutions of the form , , which implies , for some . Call . Then, by substitution in (41) we obtain

Proposition 4. Assume that is a curve in with nonconstant curvature and constant torsion which is an extremal of the energy , where and . Then, there exists a -parameter group of isometries of , , such that a suitable parametrization of the surface is a solution of (19) with .

Proof. It is easy to verify from (51) that and satisfy the Euler-Lagrange equations for the above energy ([9, 10]) for a suitable . Consider a solution of (51) (observe that (51) can be explicitly solved with the aid of Jacobi elliptic functions); then and determine a curve in which is an extremal for . Now, the vector field is a Killing field along ([9, 10]). As in previous proposition, can be extended to a Killing field on with -parameter group of isometries . Consider the surface . Then, the reparametrization satisfies , all filaments having constant torsion .

If , then and and is flat since, combining (10) and (25), we obtain that the Gaussian curvature of is given byMoreover, (51) implies that . Thus has to be either or . Flat surfaces in can be locally described as the product, with respect to the Lie group structure of , of two curves with torsions and , respectively, [13]. In order to construct explicit parametrizations solving (19) in this case, we take the complex plane, , and consider the maps defined by , where , denotes complex conjugate, , and . Endow with the semi-Riemannian metric . Then, the restriction of to the hyperquadrics , , gives two maps which are known as the standard Hopf mappings

Let be a curve in either or . Then, the complete lift (resp., ) is a Riemannian (resp., Lorentzian) flat (zero Gaussian curvature) surface in (resp., in ) which is called the Hopf cylinder on  . The covering maps and where denotes a horizontal lift of , can be used to parametrize . Assuming without loss of generality , that is, , critical curves of in or are characterized by having torsion [5, 12], and, therefore, they must be horizontal lifts via Hopf maps. Hence, we have the following.

Proposition 5. Horizontal lifts via the Hopf map of arbitrary curves of or parametrized by (54) evolve under by rigid motions and the corresponding binormal evolution surface is a Hopf cylinder of or shaped on , .

Thus, explicit parametrizations of are obtained as follows. Take an arbitrary curve in or ; then horizontal lifts of via or are given by where

Hence, one uses (54) and (55) to obtain a solution of (19). Notice that if the curve is embedded in either or , then so is in or and we have binormal Hopf cylinders with no self-intersections. Moreover, if is a closed curve then is a closed surface (a flat Hopf Tori) but the evolving filament is not closed because of the nontrivial holonomy. However, if, in addition, the area enclosed by in either or is a rational multiple of , then there are such that the horizontal lift of an -cover of is a closed filament [12].

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by MINECO-FEDER Grant MTM2014-54804-P, Gobierno Vasco Grant IT1094-16, and UPV/EHU GIU13/08, Spain. Álvaro Pámpano has been supported by Programa Predoctoral de Formación de Personal Investigador No Doctor, Gobierno Vasco, 2015.