Binormal Motion of Curves with Constant Torsion in 3-Spaces

which is a soliton equation used to model the dynamics of a thin vortex filament in an incompressible, inviscid, homogeneous, 3-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 smoothmapx : U ⊂ R →M3 r (ρ), x(s, t), verifying xt = f(󵄨󵄨󵄨󵄨󵄨∇̃xsxs󵄨󵄨󵄨󵄨󵄨 , det (xs, ∇̃xsxs, ∇̃2 xsxs) 󵄨󵄨󵄨󵄨󵄨∇̃xsxs󵄨󵄨󵄨󵄨󵄨2 )xs × ∇̃xsxs, (2) where f is a suitable smooth function, ∇̃ denotes the LeviCivita connection on M3 r (ρ), r ∈ {0, 1}, and M3 r (ρ) is a Riemannian (r = 0) or Lorentzian (r = 1) 3-space form with constant curvature ρ; that is, M3 r (ρ) is one of the following: R, the sphere S, the hyperbolic space H, the Minkowski spaceR31, the de Sitter space S 3 1, or the anti de Sitter spaceH 3 1. Under mild conditions we will see that a curve motion following (2) describes a curve γ evolving under the binormal flow, with velocity depending on curvature and torsion (19), and determines an immersed surface, Sγ, in M3 r (ρ). 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 Sγ. Alternatively, one can determine the evolution by finding a single solution, working as initial condition x(s, 0) = γ(s), and then giving a geometrical description of the binormal flow. If M3 r (ρ) = R and f ≡ 1, (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 f ≡ f(|∇̃xsxs|), 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 theGauss-Codazzi equations to construct solutions bymeans of extremal curves for curvature dependent energies and associated 1-parameter groups of isometries.


Introduction
A large class of physical systems are modelled in terms of motion of curves and surfaces in Euclidean space R 3 .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, 3-dimensional fluid [1][2][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  :  ⊂ R 2 → where  is a suitable smooth function, ∇ denotes the Levi-Civita connection on  3  (),  ∈ {0,1}, and  3  () is a Riemannian ( = 0) or Lorentzian ( = 1) 3-space form with constant curvature ; that is,  3  () is one of the following: R 3 , the sphere S 3 , the hyperbolic space H 3 , the Minkowski space R 3  1 , the de Sitter space S 3 1 , or the anti de Sitter space H 3 1 .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  3  ().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 (, 0) = (), and then giving a geometrical description of the binormal flow.
If  3  () = R 3 and  ≡ 1, (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 1-parameter groups of isometries.

Binormal Evolution Surfaces
The covariant derivative of a vector field  along a curve  will be denoted by (/)() fl ∇ ().Let () be a unit speed nongeodesic curve immersed in  3  () 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 2 or 3 and the standard Frenet frame along () is given by {, , }(), where  is chosen so that det(, , ) = 1.Then the Frenet equations define the curvature, (), and torsion, (), along () where   , 1 ≤  ≤ 3, 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.
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  3  () 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 parametrization determines 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 1.1) can be computed from the metric coefficients   .In our case, we have where 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 (s, ) denote the curvature and torsion of the curves   ().
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.(42)

Evolution with Constant Torsion
Now we study binormal evolution surfaces, whose filaments have the same constant torsion.Since  =   ∈ R, F(, ) = F((, ),   ) fl F((, )).Choose () so that Ṗ () fl / = F().Assume first  = 0. Proof.By substituting  = 0 in (40) we have (, ) = () and the metric with respect to the chosen coordinate system is  =  1  2 + 3 F2 () 2 .This means that (, ) is a warped product surface [8], and since   / = 0, we have that   (, ) = F()(, ) is a Killing field of (, ).Now, integrating (41) we get for some  ∈ R.Moreover, since  = 0, we have that (43) is the Euler-Lagrange equation for Θ() ( [9,10]) and   must be an extremal of Θ in  2  (), ∀.On the other hand, for a given field along , , the following variation formulas for V = | γ |, , 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 that along , where I = Ṗ .This means that I = Ṗ  is a Killing field along  ( [9][10][11]), but this field is precisely   = F.Now, (44) imply that the Killing vector fields along a curve  form a six-dimensional linear space.Moreover, the Lie algebra of  3  () is six-dimensional and the restriction of a Killing vector field in  3  () 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  3  () [11]; in other words,   can be extended to a Killing field on  3  () (denoted also by I).Hence, the associated 1-parameter group {  ,  ∈ R} is formed by isometries of  3  () and   = () is obtained as   = {  (()),  ∈ R}, where () = (, 0).Since   = (, ),   = Ṗ (())(, ), and ∇  (, ) = − 2 (, ) = 0, we get that (, ) does not depend on .Moreover, as fibers are orbits of a Killing field of  3  (), 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   () = (/ Ṗ ((  ))) 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 verify   (, ) =   () , from which we see that either   = 0 and () is a geodesic in  3  () or   ̸ = 0 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