# Binormal Motion of Curves with Constant Torsion in 3-Spaces

Authors: Josu Arroyo; Óscar J. Garay; Álvaro Pámpano
Journal: Advances in Mathematical Physics
 (2017)

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## 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.


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## Body

## 1. Introduction

A large class of physical systems are modelled in terms of motion of curves and surfaces in Euclidean spaceR3. A remarkable example is the so-called localized induction equation (LIE)(1)xt=xs×xss,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 smooth map x:U⊂R2→Mr3ρ, xs,t, verifying(2)xt=f∇~xsxs,det⁡xs,∇~xsxs,∇~xs2xs∇~xsxs2xs×∇~xsxs,where f is a suitable smooth function, ∇~ denotes the Levi-Civita connection on Mr3ρ, r∈0,1, and Mr3ρ is a Riemannian (r=0) or Lorentzian (r=1) 3-space form with constant curvature ρ; that is, Mr3ρ is one of the following: R3, the sphere S3, the hyperbolic space H3, the Minkowski space R13, the de Sitter space S13, or the anti de Sitter space H13.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 Mr3ρ. 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 xs,0=γs, and then giving a geometrical description of the binormal flow.IfMr3ρ=R3 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 the Gauss-Codazzi equations to construct solutions by means of extremal curves for curvature dependent energies and associated 1-parameter groups of isometries.


## 2. Preliminaries

Consider the Euclidean semispaceEνm, that is, Rm endowed with the canonical metric of index ν, denoted by ·,·, and the Levi-Civita connection, denoted by ∇¯. Then, the semi-Riemannian 3-space forms Mr3ρ (the Riemannian case, r=0 will be simply denoted by M3ρ) can be isometrically immersed in Eν4, the 4-dimensional Euclidean semispace, in a standard way [8]. The flat case, Mr3ρ=Er3,  ρ=0,  r=0,1, corresponds to either R3 or the Minkowski space R13≡L3. They can be isometrically immersed in L4=R14 endorsed with the metric(3)g=dx12+dx22+dx32-dx42in an obvious manner:(4)R3=x1,x2,x3,x4∈L4∣x4=0,L3=x1,x2,x3,x4∈L4∣x1=0.Whenρ>0, Mr3ρ correspond to the 3-sphere, S3ρ  (r=0), and the de Sitter 3-space, S13ρ  (r=1), defined by(5)Sr3ρ=x∈Er4∣x,x=1ρ,where x=x1,x2,x3,x4. Finally, for ρ<0 we obtain the hyperbolic 3-space, H3r=0, and the anti de Sitter 3-space, H13 (r=1)(6)Hr3ρ=x∈Er+14∣x,x=1ρ.The standard isometric immersions ofMr3ρ into Eν4 ([8] p. 20) will be all denoted by i and the induced metrics also by ·,·, while the Levi-Civita connections on Eν4 and Mr3ρ are denoted by ∇¯ and ∇~, respectively. As usual, the cross product of two vector fields X,Y in Mr3ρ, denoted by X  ×  Y, is defined so that X×Y,Z=det⁡X,Y,Z for any other vector field Z of Mr3ρ, where det⁡X,Y,Z stands for the determinant.Now, for a given isometric immersion of a surface,x:Nν2→Mr3ρ, ν∈0,1, we denote by ∇ the Levi-Civita connection of the immersion Nν2,x. As it is also customary, for a surface Nν2 in any 3-dimensional space form Mr3ρ, we require the first fundamental form to be nondegenerate. Take X,Y,Z,W tangent vector fields to Nν2 and choose ξ a normal vector field to Nν2 in Mr3ρ. Then theformulas of Gauss and Weingarten are, respectively, as follows [8]:(7)∇¯XY=∇~XY-ρX,Yx=∇XY+hX,Y-ρX,Yx,(8)∇~Xξ=-AξX+DX⊥ξ,where x=i∘x is the position vector, h denotes the second fundamental form of Nν2 in Mr3ρ, Aξ stands for the Weingarten map, and D⊥ denotes the connection on the normal bundle of Nν2. By using (7) and (8) and denoting by R and R~ the Riemann curvature tensors associated with ∇ and ∇~, respectively, the following relation holds:(9)R~X,YZ=ρY,ZX-X,ZY,while theequations of Gauss and Codazzi are given, respectively, by [8](10)R~X,YZ,W=RX,YZ,W-hX,W,hY,Z+hX,Z,hY,W,(11)∇hX,Y,Z=∇hY,X,Z,where ∇h is defined by(12)∇hX,Y,Z=DX⊥hY,Z-h∇XY,Z-hY,∇XZ.Now choosing an adapted local orthonormal framee1,e2,e3 in Mr3ρ such that the vectors e1, e2 are tangent to Nν2 and e3 is normal to Nν2 in Mr3ρ, and denoting by ω1,ω2,ω3 the dual frame of e1,e2,e3, theCartan connection forms are defined by(13)∇~Xei=∑jε~jωijXej,for i,j∈1,2,3, where ε~j=ej,ej is the causal character of ej. Then, ωij=-ωji and(14)hei,ej=ε3~hije3,hij=-∇~eie3,ej=ωj3ei,i
,
j
∈
1,2 [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 fieldX along a curve γ will be denoted by DX/dss≔∇~TXs. Let γs be a unit speed nongeodesic curve immersed in Mr3ρ with nonnull velocity Dγ/dss=Ts, ∀s; therefore, it is assumed to be either spacelike or timelike. If it also has nonnull acceleration DT/dss, then γs is aFrenet curve of rank 2 or 3 and the standardFrenet frame along γs is given by T,N,Bs, where B is chosen so that det⁡T,N,B=1. Then theFrenet equations define thecurvature, κs, andtorsion, τs, along γs(15)DTdss=ε2κsNs,(16)DNdss=-ε1κsTs+ε3τsBs,(17)DBdss=-ε2τsNs,where εi, 1≤i≤3, is the causal character of T, N, and B, respectively. Notice that the following relations hold:(18)T=ε1N×B,N=ε2B×T,B=ε3T×N.Curves for which both curvature and torsion are constant are calledFrenet 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 mapx:U⊂R2→Mr3ρ, xs,t, satisfying (2), we usually identify x and i∘x. Assume that theinitial conditionγs≔xs,0 is a unit speed Frenet curve of rank 2 or 3; then γts≔xs,t, which will be calledthe filament at time  t, is also unit speed parametrized ∀t. In fact, we have ∂/∂txss,t,xss,t=2∇~xsxt,xs=0, where the last equality is obtained from (2). So, since xss,0,xss,0=dγ/ds,dγ/ds=T,T=ε1, then so is ∀t; that is, (2) is a length-preserving evolution. Assuming also that ∇~xsxss,t is nonnull everywhere, the associated Frenet frame will be defined for all γt and combining (2) and (15) we obtain(19)xt=fκ,τxs×∇~xsxs=fκ,τT×DTds=ε2κfκ,τT×N=ε2ε3κfκ,τB=Fκ,τB.This means that γs evolves by the binormal flow withvelocity  Fκ,τ. We are going to suppose also that f is never zero so that U,x defines an immersed surface in Mr3ρ swept out by γs. It will be denoted by Sγ and called abinormal evolution surface (BES)with initial condition  γ and velocity  F. The curves γst≔xs,t, perpendicular to the filaments, are calledfibers of Sγ. The time variation of the Frenet frames is described in the following proposition (cf. (3.15) of [3] for surfaces in R3).Proposition 1.
LetU,x be a BES of Mr3ρ with velocity F (19). Then(20)∇~∂/∂sTNBs,t=0κ0-κ0τ0-τ0ε1Tε2Nε3Bs,t,(21)∇~∂/∂tTNBs,t=0-τFε3FsτF0ε2h22F-ε3Fs-ε2h22F0ε1Tε2Nε3Bs,t,where h22=1/κε3Fss/F-ε2τ2+ε1ε3ρ and κs,t and τs,t denote the curvature and torsion of the curves γts.Proof.
Under our assumptions, allγt 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 γts=xs,t is not a geodesic in Mr3ρ and ∇~xsxs is not null, then, for sufficiently small s, the unit Frenet normal to γts, Ns,t, is parallel to a (local) unit normal to Sγ. This means that γts are geodesics in Sγ for any t and the parametrization(22)xs,t=γtsdetermines ageodesic coordinate system with respect to which the metric ,≡g can be written as(23)g=ε1ds2+ε3F2dt2.
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 U,x. 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 gij. In our case, we have(24)Γ111=Γ112=Γ121=0,Γ122=FsF,Γ221=-ε1ε3FFs,Γ222=FtF,where subscripts s and t mean partial derivative with respect to s and t, respectively. This makes it possible to know the expression for the Levi-Civita connection of Sγ ([8], §1.4), denoted here by ∇(25)∇∂/∂s∂∂s=0,∇∂/∂s∂∂t=FsF∂∂t,∇∂/∂t∂∂t=-ε1ε3FFs∂∂s+FtF∂∂t.
As before,Ts,t,Ns,t,Bs,t represent the Frenet frames along γts, and we choose the following local adapted frame on Sγ:(26)e1=xs=T,e2=xtF=B,e3=ξ=-ε2N,where ξ is the unit normal to Sγ (locally defined). Then, combining (7), (8), (13), (14), and (15)–(17), one gets(27)ω12e1=0,ω12e2=ε3FsF,ω13e1=h11=-ε2κ,ω13e2=ω23e1=h12=ε2τ,ω23e2≔h22,
whereκs,t and τs,t denote the curvature and torsion of the curves γts.
The second fundamental form can be considered as a quadratic form given byhX≔AξX,X; therefore, we obtain from (27) that(28)h=-ε2κds2+2ε2τFdsdt+F2h22dt2,with respect to the parametrization (22), where h22 is the coefficient of the second fundamental form ([8], §2.3) of Sγ in Mr3ρ given by h22≔-ε2∇~BB,N. Since ∇ is determined by g, h22 can be computed with the aid of (25) and the Gauss formula (7) giving(29)h22=1κε3FssF-ε2τ2+ε1ε3ρ.
Using again the Gauss and Weingarten formulas (7) and (8), it can be shown that x:U→Mr3ρ→Eν4 satisfies the following PDE system:(30)xss=-κFxs×xt-ε1ρx,(31)xts=FsFxt-τFκxss-ε1τFκρx,(32)xtt=-ε1ε3FFsxs-ε2h22F2κxss-ε1ε2h22κ+ε3F2ρx+FtFxt.
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 have(33)DTdt=∇~∂/∂tT=-ε2FτN+FsB.So, differentiating xt=FB and using once more Gauss and Weingarten formulas, we have(34)xtt=FtB+F∇~∂/∂tB-ε3ρFx,
which combined with (32) gives(35)DBdt=∇~∂/∂tB=-ε1ε3FsT-h22FN.Finally, from (33), (35), and the cross product relations (18), one has(36)DNdt=∇~∂/∂tN=ε1τFT+ε2ε3h22FB,
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 computation(37)ε2ε3FssF=ε2κh22+τ2-ε1ε2ε3ρ,(38)κt=-2Fsτ-τsF,(39)τt=ε1ε3κFs+ε2h22Fs.Notice 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 U,x boil down to(40)κt=-2Fsτ-τsF,(41)τt=ε1ε3κFs+ε2Fκε3FssF-ε2τ2+ε1ε3ρs.Observe that ifF=κ, (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 κs,t, τs,t, and Fs,t, smoothly defined on a connected domain U and satisfying (40) and (41), there exists a solution of (30)–(32) (and, consequently, of (20)-(21)) determining a smooth isometric immersion U,x (unique up to rigid motions, if U is simply connected) of a surface in Mr3ρ whose metric and the second fundamental form are given, respectively, by g=ε1ds2+ε3F2dt2 and ε2h=-κds2+2τFdsdt+ε2F2h22dt2, where h22 is obtained from (29).Computing the Christoffel symbols from the metric coefficients for such an immersion,U,x, we see that γts are arc-length parametrized geodesics ∀t. Then, a combination of the Gauss formula (7) and (15)–(17) for the Frenet frame along the coordinate curves γts, xs=Ts,t,Ns,t,Bs,t, shows that the unit Frenet normals Ns,t are perpendicular to the surface U,x. Hence, xt=λs,tBs,t, but then the second coefficient of g (23) implies that Fs,t=λs,t and U,x is a solution of (19). Since U,x is foliated by geodesics γts having κts≔κs,t and τts≔τs,t as curvature and torsion, respectively, the immersion itself, x, is geometrically determined by κts≔κs,t and τts≔τs,t, because, from the fundamental theorem of curves, for any fixed t, there exists a unique curve γts (up to congruences and causal character of the Frenet frame) having κts and τts as curvature and torsion. Then, smooth assembling of these curves γts,  t∈J, would give x. 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.
LetU,x be a binormal evolution surface in Mr3ρ with velocity F and assume that s∈0,1 and all filament curves γts=xs,t are C4-closed in 0,1. If Fκ,τ=Fκ, then ∫01τds‍ and ∫01‍Pκ+ητds, with dP/dκ=Fκ, η∈R, are independent of t. Furthermore, if Fκ is constant, then also ∫01κds‍ is invariant. Finally, if Fκ,τ=nκ+mτ, then ∫o1mκ-ε1ε3nτds‍ does not depend on t.Proof.
Only first part is proved since the others are similar. IfFκ,τ=Fκ, then, the invariance ∫01‍τds is a direct consequence of κFs=κF-Ps, because using (41) we have d/dt∫01τds‍=∫01τtds‍=0. Now, from (40)(42)ddt∫01Pκ-ητds=∫01κtFds-η∫01τtds=∫01F2τsds=0.


## 4. Evolution with Constant Torsion

Now we study binormal evolution surfaces, whose filaments have the same constant torsion. Sinceτ=τo∈R, Fs,t=Fκs,t,τo≔F~κs,t. Choose Pu so that P˙u≔dP/du=F~u. Assume first τ=0.Proposition 3.
LetSγ be a binormal evolution surface, whose filaments satisfy τ=0. Then γ is extremal for the energy Θγ≔∫γ‍Pκ+λds and Sγ=ϕtγ,t∈R, where ϕt,t∈R is a 1-parameter group of isometries of Mr3ρ. Moreover, the fibers of Sγ have constant curvature and zero torsion (if they are not geodesics) in Mr3ρ. In particular, if ρ=0, Sγ are either ruled surfaces or rotational surfaces.Proof.
By substitutingτ=0 in (40) we have κs,t=κs and the metric with respect to the chosen coordinate system is g=ε1ds2+ε3F~2sdt2. This means that U,x is awarped product surface [8], and since ∂gij/∂t=0, we have that xts,t=F~sBs,t is a Killing field of U,x. Now, integrating (41) we get(43)0=P˙ss+ε1ε2P˙κ2+ε2ρ-ε1ε2κP+λ,for some λ∈R. Moreover, since τ=0, we have that (43) is the Euler-Lagrange equation for Θγ ([9, 10]) and γt must be an extremal of Θ in Mr2ρ, ∀t. On the other hand, for a given field along γ, W, the following variation formulas for v=γ˙, κ, and τ can be obtained using standard computations and the Frenet equations (see, [9, 10])(44)Wv=ε1v∇~TW,T,Wκ=∇~T2W,N-2ε1κ∇~TW,T+ε1ρW,N,Wτ=ε21κ∇~T2W+ε1ρW,Bs-ε1τ∇~TW,T+ε1κ∇~TW,B,
where the subscripts denotes differentiation with respect to the arc-length. Now, combining (43), (44), and the Frenet equations, one can see that(45)Iv=Iκ=Iτ=0,along γ, where I=P˙B. This means that I=P˙B is a Killing field along γ ([9–11]), but this field is precisely xt=F~B. Now, (44) imply that the Killing vector fields along a curve γ form a six-dimensional linear space. Moreover, the Lie algebra of Mr3ρ is six-dimensional and the restriction of a Killing vector field in Mr3ρ to any curve γ gives a Killing vector field along γ. Hence, every Killing vector field along a curve, γ, is the restriction to γ of a Killing vector field of Mr3ρ [11]; in other words, xt can be extended to a Killing field on Mr3ρ (denoted also by I). Hence, the associated 1-parameter group ϕt,t∈R is formed by isometries of Mr3ρ and Sγ=xU is obtained as Sγ=ϕtγs,t∈R, where γs=xs,0. Since xs=Ts,t, xt=P˙κsBs,t, and ∇~xsBs,t=-ε2τNs,t=0, we get that Bs,t does not depend on s. Moreover, as fibers are orbits of a Killing field of Mr3ρ, they have constant curvature. Now, for any so take an arc-length parametrization, δt, of the fiber of Sγ through so. With the subscript δ denoting the geometric elements associated with the curve δ, we have Tδt=Bt/P˙κso and, using the last equation of (21), we obtain(46)ε2δκδNδ=-ε1ε3P˙sP˙T-h22N,
ifδ has nonnull acceleration. Thus, differentiating (46) with respect to t and using again (21), we have that δt must verify(47)κδs,t=κδs,κδτδ=0,ε2δκδ2=ε1P˙s2P˙2+ε2h222,from which we see that either κδ=0 and δt is a geodesic in Mr3ρ or κδ≠0 and δt is a planar circle.
On the other hand, ifδ has null acceleration and is not a geodesic, then we can consider the following frame along γ. Define Nδt as the lightlike field on δ given by DTδ/dtt=Nδt and denote by Bδt the only lightlike vector such that Nδ,Bδ=1 and Tδ,Bδ=0. In this case, we have the following equations:(48)DTδdtt=Nδt,DNδdtt=τδtNδt,DNδdtt=-Tt-τδtBδt,for certain function τδt which will be also called thetorsion of γ (here, the “curvature” is considered to be 1). Then, from the second equation of (21), it is clear that τδt=0.
Finally, we restrict ourselves to flat ambient spacesR3 or L3. For simplicity, we take first Mr3ρ=R3. If Pκ=νκ, ν∈R, then any planar curve γ is critical for Θ [12] and Sγ must be a right cylinder shaped on γ. Assume then that Pκ≠νκ, ν∈R. Then the Killing field along γ, I, can be written as I=P˙B=λ1γ×V+λ2V, for some λ1,λ2∈R, λ1≠0, and a constant vector V in R3. By scalar multiplication of V with the covariant derivative of I=P˙B=λ1γ×V+λ2V along γ, we obtain λ2=0 and, then, Sγ is a rotational surface in R3 with profile curve γ. These facts can be extended to the Minkowski space L3 by similar arguments.Moreover, ifPκ≠νκ, ν∈R, it is not difficult to see that, identifying the plane containing γ and R2, choosing a coordinate system containing V, and using I=P˙B, it is possible to find a coordinate system in R2 where γ=γ1,γ2 and γ1=ηP˙, for some constant η∈R. This also works in the Minkowski space L3. In particular, take an extremal curve for ∫γ‍Pκ+λds either in R2 or in L2 and choose a coordinate system where γ=γ1,γ2 and γ1=ηP˙=ηF~, for some constant η∈R. Assuming η≠0, then(49)xs,t=ηF~sin⁡t,ηF~cos⁡t,as,ifε1δε2δ=1,xs,t=ηF~sinh⁡t,as,ηF~cosh⁡t,ifε1δ=-1,xs,t=ηF~cosh⁡t,as,ηF~sinh⁡t,ifε2δ=-1,where as verifies that ε2δηF~′2+ε3δa′s2=ε1 are rotation surfaces with planar filaments evolving by (19). If the extremal curve lies in the Minkowski plane L2, then choosing the same coordinate system as before, one could also construct a surface with planar filaments evolving by (19) by rotating γ around a lightlike axis. In fact, suppose without loss of generality that the lightlike axis is determined by ∂/∂x3+∂/∂x4; then, the parametrization of Sγ is given by(50)xs,t=ηF~+as-ηF~t22,as-ηF~t,as+as-ηF~t22,whereas verifies ηF~′2-a′s2=ε1. In this case, fibers are spacelike curves with null acceleration.We remark that a periodic solutionκs of (43) does not determine a closed filament necessarily, and closure conditions have to be derived. For instance, if κs is a periodic solution of (19) with period ρ and γs is the corresponding curve in R2, then integrating (43) we have d=P˙s2+κP˙-P-λ2, for some d∈R. Choosing a coordinate system in R2 where γ=γ1,γ2 and γ1=1/dP˙, we see that γ1 is also periodic. Using the fact that γ is arc-length-parametrized, we have γ2′s=1/dκP˙-P-λ. Hence, γ closes up, if and only if ∫oρκP˙-P-λ‍=0.Take nowτo≠0. If κs,t is also constant, filaments are Frenet helices. The unit binormal B is a Killing vector field on Frenet helices; hence, their evolution under the B-flow satisfies xt=B, so we assume κs,t is not constant. Then (40) suggests studying travelling wave solutions of the form κs-ε1ε3μt, μ∈R, which implies F~κs,t=ε1ε3μ/2τ0κ+λ, for some λ∈R. Call ι=s-ε1ε3μt. Then, by substitution in (41) we obtain(51)0=ε1ε2μ2τoκιι+μ4τoκ3+-1rμ2τoκε1ε3ρ-ε2τo2+λ-1rρ-τo2.Proposition 4.
Assume thatγι is a curve in Mr3ρ with nonconstant curvature and constant torsion τ=τo≠0 which is an extremal of the energy Θ~γ=∫γε1ε3μ/4τoκ2+λκ+μτ+η‍, where μ≠0 and λ,η∈R. Then, there exists a 1-parameter group of isometries of Mr3ρ, ϕt,t∈R, such that a suitable parametrization of the surface Sγ≔ϕtγι is a solution of (19) with F~κs,t=ε1ε3μ/2τ0κ+λ.Proof.
It is easy to verify from (51) that κι and τ=τo satisfy the Euler-Lagrange equations for the above energy Θ~ ([9, 10]) for a suitable η∈R. Consider κι a solution of (51) (observe that (51) can be explicitly solved with the aid of Jacobi elliptic functions); then κι and τ=τo determine a curve γι in Mr3ρ which is an extremal for Θ~. Now, the vector field Iι≔ε1ε3μTι+F~ιBι is a Killing field along γι ([9, 10]). As in previous proposition, I can be extended to a Killing field on Mr3ρ with 1-parameter group of isometries ϕt,t∈R. Consider the surface Sγ≔yι,t=ϕtγι,t∈R. Then, the reparametrization xs,t≔ys-ε1ε3μt,t satisfies xt=ε1ε3μ/2τ0κ+λB, all filaments having constant torsion τ=τo.Ifμ=0, then κs,t=κs and F~κs=λ and Sγ is flat since, combining (10) and (25), we obtain that the Gaussian curvature of Sγ is given by(52)K≔R∂/∂s,∂/∂t,∂/∂t,∂/∂sg∂/∂s,∂/∂sg∂/∂t,∂/∂t-g∂/∂s,∂/∂t2=-ε1FssF=0.Moreover, (51) implies that ρ=-1rτo2. Thus Mr3ρ has to be either S3ρ or H13ρ. Flat surfaces in S3ρ can be locally described as the product, with respect to the Lie group structure of S3ρ, of two curves with torsions ρ and -ρ, respectively, [13]. In order to construct explicit parametrizations solving (19) in this case, we take the complex plane, C, and consider the maps πε:C2→C2 defined by πεz1,z2=1/2cz¯1z1-εz¯2z2,2z2z¯1, where zi∈C,i∈1,2, z¯i denotes complex conjugate, ε=±1, and c∈R. Endow C2 with the semi-Riemannian metric z,w=Real⁡z1w¯1+εz2w¯2. Then, the restriction of πε to the hyperquadrics z,z=εc2, ε=±1, gives two maps which are known as thestandard Hopf mappings(53)π+:S31c2⟶S24c2,π-:H13-1c2⟶H2-4c2.Letγ be a curve in either S24/c2 or H2-4/c2. Then, the complete lift Tγ+=π+-1γ (resp., Tγ-=π--1γ) is a Riemannian (resp., Lorentzian) flat (zero Gaussian curvature) surface in S31/c2 (resp., in H13-1/c2) which is called theHopf cylinder on  γ. The covering maps Ψ+:R2→Tγ+ and Ψ-:L2→Tγ-(54)Ψ±t,s=eitγ¯s,where γ¯s denotes ahorizontal lift of γ, can be used to parametrize Tγ±. Assuming without loss of generality c=1, that is, ρ=±1, critical curves of ∫γκ‍ in S31 or H13-1 are characterized by having torsion τ2=1 [5, 12], and, therefore, they must be horizontal lifts via Hopf maps. Hence, we have the following.Proposition 5.
Horizontal lifts via the Hopf mapπ± of arbitrary curves γ of S24 or H2-4 parametrized by (54) evolve under xt=B by rigid motions and the corresponding binormal evolution surface is a Hopf cylinder of S31 or H13-1 shaped on γ, Tγ±.Thus, explicit parametrizations ofTγ± are obtained as follows. Take an arbitrary curve γs=A1s,0,A2s,A3s in S24 or H2-4; then horizontal lifts of γ via π+ or π- are given by (55)γ¯s=Mscos⁡αs,Mssin⁡αs,Dscos⁡αs-Pssin⁡αs,Dssin⁡αs+Pscos⁡αs,where(56)Ms=2A1s+12,Ds=2A2s2A1s+1,Ps=2A3s2A1s+1,αs=±2∫A3sA2′s-A2sA3′s2A1s+1.Hence, one uses (54) and (55) to obtain a solution of (19). Notice that if the curve γ is embedded in either S24 or H2-4, then so is Tγ± in S31 or H13-1 and we have binormal Hopf cylinders with no self-intersections. Moreover, if γ is a closed curve then Tγ± is a closed surface (a flat Hopf Tori) but the evolving filament γ¯s is not closed because of the nontrivial holonomy. However, if, in addition, the area enclosed by γs in either S24 or H2-4 is a rational multiple of π, then there are m∈Z such that the horizontal lift of an m-cover of γs is a closed filament [12].



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*Source: 7075831-2017-06-29.xml*