# Hypersurface Constrained Elasticae in Lorentzian Space Forms

Authors: Óscar J. Garay; Álvaro Pámpano; Changhwa Woo
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

We study geodesics in hypersurfaces of a Lorentzian space formM1n+1(c), which are critical curves of the M1n+1(c)-bending energy functional, for variations constrained to lie on the hypersurface. We characterize critical geodesics showing that they live fully immersed in a totally geodesic M13(c) and that they must be of three different types. Finally, we consider the classification of surfaces in the Minkowski 3-space foliated by critical geodesics.


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## Body

## 1. Introduction

Following a classical D. Bernoulli’s model, a curve immersed in a Riemannian manifoldγ:I→Mn is called anelastic curve (or simply anelastica) if it is a minimum, or, more generally, a critical point, of thebending energy∫γkg2, where kg denotes thegeodesic curvature of γ in Mn. The study of elasticae is a classical variational problem initiated in 1691 when J. Bernoulli proposed determining the final shape of a flexible rod. If n=2, the problem of elastic curves in surfaces has a long history, but it is really well understood only when M2 is a real 2-space form. In fact, Euler published in 1744 his classification of the planar elastic curves [1], and, much more recently, Langer and Singer have classified the closed elastic curves in the 2-sphere and in the hyperbolic plane [2], but, in general, little is known about elastic curves in surfaces with nonconstant curvature. Since 1691, elastica related problems have shown remarkable applications to many different fields having drawn the attention of a wide range of scientists who have developed different approaches to deal with them (for more details on this subject see, for instance, [3] and the references therein). Elastica related problems have also been considered in pseudo-Riemannian ambient spaces (see, e.g., [4–6]).On the other hand, ifγ:I→Mn⊂Mm(c) is a curve on a submanifold Mn immersed in a real m-space form of constant curvature c, Mm(c), one may wish to analyze the critical curves for the bending energy of the curve in Mm(c), but for variations of γ constrained to lie on the submanifold. In this paper, this problem will be referred to as theelastica constrained problem. The constrained problem was first considered by Santaló in the context of Euclidean surfaces, c=0 [7]. In particular, he obtained the Euler-Lagrange equation of ∫κ2, with κ being the curvature in R3 of a curve γ:I→M2⊂R3, for variations of γ constrained to lie in M2 with prescribed first-order boundary data. Different versions of this problem for surfaces in 3-space forms for a variety of curvature energies and boundary conditions have been considered in [8–11]. In particular, in the aforementioned works, the Euler-Lagrange equation for constrained elasticae has been computed in invariant form; however, it is a long complicated equation difficult to deal with even for curves immersed in surfaces. On the other hand, it is known that every geodesic of pseudo-Riemannian manifold is an elastica, but, in contrast with this fact, not every geodesic of a submanifold is a constrained elastica. Thus, it makes sense to study geodesics of submanifolds which are critical for the constrained problem, separately.This paper is devoted to study geodesics of hypersurfacesMrn,r=0,1, in a Lorentzian space form M1n+1(c), which are critical for the constrained problem. In Section 2, we review a few basic facts which will be needed later. In Section 3, we compute the first variation formula and characterize the geodesics of Mrn, which are critical for the elastic energy of M1n+1(c) for variations constrained to lie on the hypersurface. In Proposition 3, we obtain three different types of possible critical geodesics for the constrained problem which are characterized in terms of their Frenet curvatures. We also prove that they must live in a totally geodesic M13(c). Then, in Section 4, we restrict ourselves to the flat Minkowski space L3 and study surfaces which are foliated by critical geodesics of each type. Whilst surfaces foliated by first-type critical geodesics are basically ruled surfaces, in Proposition 4 we completely classify (locally) surfaces foliated by geodesics of the second type (planar geodesics). At the end of this section, we introduce Hashimoto surfaces in L3 with a rank 3 Frenet curve as initial condition and show a few properties that they share with their Riemannian counterparts. As a consequence, we obtain that elasticae of the Minkowski 3-space evolve by rigid motions; that is, they move without change of shape under the binormal flow and produce surfaces which are foliated by critical geodesics of the third type (Proposition 5).


## 2. Preliminaries

LetM1n be a Lorentzian manifold with metric , and Levi-Civita connection ∇. If γ:I→M1n is a smooth immersed curve in M1n, γ′(t) will represent its velocity vector dγ(t)/dt and the covariant derivative of a vector field X(t) along γ will be denoted by DX(t)/dt. A C1 immersed curve in Lorentzian manifold isspacelike (resp.,timelike; resp.,light-like) if 〈γ′(t),γ′(t)〉>0, ∀t∈I (resp., 〈γ′(t),γ′(t)〉<0, ∀t∈I; resp., 〈γ′(t),γ′(t)〉=0, ∀t∈I). Of course, there exist curves whose causal character changes as t moves along the parameter interval, but this kind of curves will not be considered here. A nonnull curve can be parametrized by the arc-length and this natural parameter is calledproper time and usually denoted by s.For a nonnull immersed curve parametrized by arc-lengths, thefirst Frenet curvature, or simplythe curvature, is defined as κ1(s)=ε2〈Dγ′s/ds,Dγ′s/ds〉, where ε2 denotes the causal character of Dγ′(s)/ds. A geodesic is a constant speed curve whose tangent vector is parallel propagated along itself, that is, a curve whose tangent, γ′(s)=T(s), satisfies the equation DT(s)/ds=0. Obviously, geodesics have zero curvatures. In this paper, geodesics will be calledFrenet curve of rank1. An immersed curve in a Lorentzian manifold γ:I→M1n is called aFrenet curve of rankm, 2≤m≤n, if m is the highest integer for which there exists an orthonormal frame defined along γ, {e1(s)=γ′(s),e2(s),…,em(s)}, and nonnegative smooth functions on γ, κi(s), s∈I, 1≤i≤m-1 (Frenet curvatures), such that the following equations are satisfied (Frenet-Serret equations):(1)Dγ′sds=De1sds=ε2κ1se2s,⋮Dehsds=-εh-1κh-1seh-1s+εh+1κhseh+1s,h∈2,3,…,m-1,⋮Demsds=-εm-1κm-1sem-1s,whereεi≔eis,eis denotes the causal character of ei(s), i∈1,…,m. For a Frenet curve of rank m<n, the Frenet curvatures of index higher than m-1 are considered to be zero, κi(s)=0, s∈I, m≤i≤n-1.A complete, connected, simply connected, Lorentzian manifold with constant sectional curvaturec is called aLorentzian space formM1n+p(c). The fundamental theorem for Frenet curves of rank m tells us that, in a Lorentzian space form, the causal characters of the Frenet frame and the Frenet curvatures k1,…,km-1 completely determine the curve up to isometries. Moreover, given functions k1,…,km-1 we can always construct a spacelike (resp., timelike) Frenet curve, parametrized by the arc-length, whose curvature functions are precisely the k1,…,km-1. Then, any local geometrical scalar defined along Frenet curves can always be expressed as a function of their curvatures and derivatives.Now, letϕ:Mrn→M1n+p(c), r∈{0,1}, be a semi-Riemannian manifold isometrically immersed in a Lorentzian space form M1n+p(c). For a given x∈Mrn, thefirst normal spaceN1(x) is defined to be the subspace of the normal space spanned by the vector valued second fundamental form of the immersion. A normal subbundle N is calledparallel if, for each section ξ of N and each tangent vector X∈TMrn, the covariant derivative of ξ in the direction of X, with respect to the normal connection, remains in N. The following result is basically known.Proposition 1.
Letϕ:Mrn→M1n+p(c), r∈{0,1}, be a semi-Riemannian manifold isometrically immersed in Lorentzian space form M1n+p(c). Assume that Mrn,ϕ has a parallel normal subbundle N2 of rank q<p containing the first normal space N1. Then, there exists a totally geodesic submanifold En+q embedded in M1n+p(c) such that ϕMrn⊂En+q.Remark 2.
Ifc=0, the result can be found in [12] for N1=N2 (see also [13], Theorem 2.6). If c=-1, a proof is given in [14]. A general proof for any c can be made by adapting to the Lorentzian case and the arguments of [13], Theorem 2.6, and of [15], Theorem 4.1. When the ambient manifold is Riemannian, the above result is due to Erbacher [16].


## 3. Hypersurface Constrained Elasticae

In this section, we assume that all our curves are nonnull with nonnull acceleration. In other words, we assume thatγ′(t) and Dγ′/dt are not light-like vectors along the curve. We use γ(s) for the arc-length reparametrization of γ(t) and in such a case the velocity vector is denoted by γ˙(s)=dγ(s)/ds. Spacelike (resp., timelike) geodesics can be characterized as those constant speed immersed curves which are minimizers (resp., maximizers) of thelength functionalΥ(γ)≔∫γε1γ′,γ′, with ε1 being the causal character of the curve, among spacelike (resp., timelike) curves joining the same end points.Elastic curves or, simply,elasticae are defined as those curves which are critical for thebending energy functional:(2)Fγ≔∫γε2Dγ˙ds,Dγ˙ds+λds,where ε2=1 or -1 depending on whether Dγ˙/ds is spacelike or timelike, s is the arc-length parameter, γ˙ represents derivative with respect to s, and λ is a real constant (F is supposed to act on a space of curves satisfying suitable initial and first-order boundary conditions, e.g., the space of curves Ω defined in (4)). Clearly, geodesics of M1n are elasticae. Moreover, elastic Frenet curves of Lorentzian space forms M1n(c) are known to lie fully in a totally geodesic submanifold of dimension at most 3, M13(c) (for details, see [4]).Now, letϕ:Mrn-1→M1n(c) be a semi-Riemannian hypersurface of index r isometrically immersed in a Lorentzian space form M1n(c). In this case, Mrn-1 can be either Riemannian, r=0, or Lorentzian, r=1. Assume that γ:I=[0,1]→Mrn-1 is a smooth immersed curve contained in the hypersurface. We are interested in those curves γ of the hypersurface which are critical points of the bending energy (2) for variations contained in Mrn-1. For simplicity, along this work, this problem will be referred to asthe hypersurface constrained problem forF or Mrn-1-constrained problem forF. In contrast with what happens in the unconstrained problem, it turns out that a geodesic of Mrn-1 is not necessarily a critical curve for the constrained problem. We first want to characterize geodesics of Mrn-1 which are Mrn-1-constrained critical.In order to derive first variation formulas forF, we will use the following standard terminology (see [2, 5, 6] for details). For a nonnull immersed curve γ:I→Mrn-1, with I≔[0,1], we denote v(t)=ε1γ′,γ′(t) and by T=T(t) its unit tangent vector. Take a variation of γ, Γ=Γ(t,w):[0,1]×(-η,η)→Mrn-1 with Γ(t,0)=γ(t). Associated with this variation, we have thevariation vector fieldW=Wt≔∂Γ/∂w(t,0) along the curve γ(t). We also write V=V(t,w)=∂Γ/∂t(t,w), W=W(t,w), v=v(t,w), T=T(t,w), and so forth, with the obvious meanings. Let s denote the arc-length, and put Vs,w, W(s,w), and so forth, for the corresponding reparametrizations. The following formulas can be computed as in [2, 5](3)W,T=-ε1∇TW,TT,Wv=ε1∇TW,Tv,Wκ2=2ε2∇T2W,∇TT-4ε1κ2∇TW,T+2ε2RW,TT,∇TT,where the Riemannian curvature tensor R is defined by R(X,Y)Z≔∇X∇YZ-∇Y∇XZ-∇[X,Y]Z.Choose two arbitrary pointspi∈Mrn-1 and vectors vi∈TpiMrn-1, i∈{0,1}, and consider the space of curves(4)Ω=β:I⟶Mrn-1∣dβdt,dβdtt≠0,βi=pi,dβdti=vi,i∈0,1,where dβ/dt(t) denotes the derivative with respect to the parameter t∈I. We wish now to analyze the variational problem associated with energy (2) acting on Ω.Proposition 3.
Letγ:I→Mrn-1⊂M1n(c) be a Frenet curve of rank m which is geodesic of Mrn-1. Assume that F is acting on Ω. Then, γ is a critical point of F (i.e., for the hypersurface constrained problem), if and only if one of the following cases occurs: (1)
Rank ofγ is 1; that is, γ is a geodesic of M1n(c).(2)
Rank ofγ is 2. That is, the torsion of γ vanishes, κ2=0.(3)
γ is a Frenet curve of rank 3 satisfying(5)κ12κ2=d,where d∈R is a constant and κ1, κ2 are the two first Frenet curvatures of γ in M1n(c). Moreover, in all the above cases γ lies fully in a totally geodesic submanifold Em⊂M1n(c) of dimension m=rankγ, 1≤m≤3.Proof.
Letγ∈Ω and take a variation Γ=Γ(t,w):[0,1]×(-ϵ,ϵ)→Mrn-1 of γ, Γ(t,0)=γ(t) by curves in Ω of the same causal character. Now, we use standard arguments, the above formulas (3), and integration by parts (see [2, 5]), to obtain the first variation formula of F along γ in the direction of W:(6)δFγW=∫γEγ,Wds+Bγ,W01,where E(γ) and B(γ,W) stand for the Euler-Lagrange and Boundary operators, respectively, which are given by(7)Eγ=2ε2D3Tds3+3ε1Dκ2Tds+2ε2RDTds,TT-λε1DTds,(8)Bγ,W=2ε2DWds,DTds-3ε1κ2T,W-2ε2W,D2Tds2+λW,T.We see that the initial and boundary conditions of the variation imply that the boundary term B(γ,W)01 vanishes. Moreover, in a Lorentzian space form, the curvature tensor is given by (9)RX,YZ=cY,ZX-X,ZY,so RDT/ds,TT=cε1DT/ds and (7) becomes(10)Eγ=2ε2D3Tds3+3ε1Dκ2Tds+ε12cε2-λDTds.Since γ⊂Mrn-1 and we are taking variations in Mrn-1, the variation field W is tangent to Mrn-1 along γ. So only the tangential part of E affects the first variation formula (6) and γ is a critical point of F, if and only if(11)tan⁡Eγ=0,where tan⁡ denotes tangential projection on Mrn-1.
Now, if rank ofγ is 1, then it is a geodesic of M1n(c) and (11) trivially holds. Observe that if γ is a geodesic of M1n(c), then it is a critical curve of the bending energy for any variation and so is the case for constrained variations.
If rank ofγ is 2, then(12)Dγ′tdt=De1tdt=ε2κ1te2t,(13)De2tdt=-ε1κ1te1t.Using (10), one sees that E(γ)=ϕe2, for a certain function ϕ along the curve. But e2 is normal to Mrn-1, since γ is a geodesic of the hypersurface, what means that tan⁡Eγ=0 and the curve is critical. On the other hand, the first normal space N1 along γ is spanned also by e2 and (13) shows that N1 is a parallel normal subbundle of dimension 1. Hence, applying Proposition 1, we have that γ lies fully in a totally geodesic submanifold E2⊂M1n(c) of dimension 2.
If rank ofγ is 3, then the Frenet equations reduce to(14)Dγ′tdt=De1tdt=ε2κ1te2t,De2tdt=-ε1κ1te1t+ε3κ2te3t,De3tdt=-ε2κ2te2t.Using (10), one sees that E(γ)=ϕ2e2+ϕ3e3, for certain functions ϕi,i∈{2,3} along the curve. Again, e2 is normal to Mrn-1, because γ is a geodesic of the hypersurface, what means that tan⁡Eγ=ϕ3e3. Hence, the curve is critical, if and only if ϕ3=0. A computation gives ϕ3=4ε3κ1′κ2+2ε3κ1κ2′, from where we obtain that κ12κ2 must be constant along the curve. Moreover, if we consider the 2-dimensional normal bundle N=span{e2,e3}, we see that it contains the first normal space along γ, N1=span{e2}. In addition, the two last equations of (14) tell us that N is parallel, so, applying again Proposition 1, we have that γ lies fully in a totally geodesic submanifold E3⊂M1n(c) of dimension 3.
Finally, if rank ofγ were 4, then, by using a similar argument, criticality of γ would imply also that E(γ)=ϕ2e2+ϕ3e3+ϕ4e4 should vanish. But a computation involving the Frenet equations and (10) would give ϕ4=κ1κ2κ3=0, which contradicts that the rank is 4. An analogous argument works for any higher rank.In particular, restricting ourselves to the flat ambient space case, we know from Proposition3 that geodesics of a surface Mr2 immersed in the Minkowski 3-space L3, which are critical for the surface constrained problem, must fall under one of the three cases described there. We want to study surfaces in L3 foliated by such critical geodesics. The main goal of the next section is to determine all surfaces of L3 locally foliated by critical geodesics of type 2 of Proposition 3. Then, in the final part of the section we will give a method to construct surfaces of L3 locally foliated by critical geodesics of type 3 of Proposition 3.


## 4. Surfaces inL3 Foliated by Surface Constrained Critical Geodesics

Consider the Minkowski3-space L3, that is, the flat Lorentzian 3-space R3 equipped with the metric(15)go=-dx12+dx22+dx32,where (x1,x2,x3) is the standard rectangular coordinate system. The standard metric (15) will be denoted by 〈·,·〉. As usual, the cross product of two vector fields X, Y in L3, denoted by X×Y, is defined so that X×Y,Z=det⁡X,Y,Z for any other vector field Z of L3.Letγ be a unit speed nongeodesic curve contained in L3 with nonnull velocity γ˙=T. If it also has nonnull acceleration Dγ˙/ds, then γ is a Frenet curve of rank 2 or 3 and the classical standard Frenet frame along γ is given by {T=γ˙,N=ε2/κ1∇~TT,B}, and B is chosen so that det⁡T,N,B=1. From now on, the first and second Frenet curvatures κ1,κ2 will be denoted by {κ,τ} and will be referred to as thecurvature andtorsion of γ in L3, respectively. Then, theFrenet equations (1) can be written as(16)DTds=∇~TT=ε2κN,DNds=∇~TN=-ε1κT+ε3τB,DBds=∇~TB=-ε2τN,where εi, 1≤i≤3, denotes the causal character of T, N, and B, respectively, and the following relations hold:(17)T=ε1N×B,N=ε2B×T,B=ε3T×N.Notice that even if the rank of γ is 2, the binormal B=ε3T×N is still well defined and above formulas (16) still make sense when τ=0.Ifγ had null acceleration Dγ˙/ds, then we would consider the following frame along γ. Take N(s)=∇~TT and denote by B(s) the only light-like vector such that 〈N,B〉=1 and 〈T,B〉=0. Again, the vectors Ns,Bs are referred to as the unit normal and binormal vectors of γ, respectively. In this case, the “Frenet” equations are(18)DTds=∇~TT=N,DNds=∇~TN=τN,DBds=∇~TB=-T-τB,for a certain function τ(s) which will be also calledtorsion. There is no definition for curvature in this case.On the other hand, aruled surfaceS in 3-space L3 is defined by the property that it admits a parametrization x(s,t)=α(s)+tX(s), where α(s) is a connected piece of a regular curve and X(s) is a nowhere vanishing vector field along the curve. Thus, rulings (s=constant) of S are geodesics of L3 and ruled surfaces are examples of surfaces foliated by curves of the first type of Proposition 3.Also,rotation surfaces provide us with surfaces of L3 locally foliated by critical geodesics of type 2. By aLorentzian rotation around an axis is meant Lorentzian transformation leaving a straight line (the axis) pointwise fixed. Rotation surfaces are those surfaces in L3 which are invariant by the 1-parameter group of the Lorentzian isometries which leave a straight line L (the axis of revolution) pointwise fixed. There are three types of rotation surfaces, depending on the causal character of the axis (timelike, spacelike, or null) [17]. In all three cases, meridians of the surface (congruent copies of the generating curve) are planar geodesics so we have infinitely many examples of both spacelike and timelike surfaces foliated by geodesics which are surface constrained elastica with τ=0 (second type of Proposition 3).Now, for a given ruled surfacex(s,t)=α(s)+tX(s), the curve α(s) is called abase curve and X(s) adirector curve. In particular, the ruled surface is said to becylindrical if the director curve X(s)=V is constant andnoncylindrical otherwise. If, in addition, V is perpendicular to α′(s), then x(s,t)=α(s)+tV is called aright cylinder on α(s). Now, assume that the base curve α(s) is a null (light-like) curve in L3 with Cartan frame {A,X,C}; then, x(s,t)=α(s)+tX(s) is a Lorentz surface which is called a null X-scroll over α(s), [18]. On the other hand, a Frenet curve is called aFrenet helix if it has constant Frenet curvatures. A Frenet helix is said to be degenerate if its axis is null and will be called nondegenerate otherwise. Nondegenerate Frenet helices are geodesics in right cylinders shaped over curves with constant curvature, Cα,V, while degenerate Frenet helices are geodesics in flat X-scrolls, Sα,X [5]. Hence, in addition to the foliation by geodesics of L3, these Cα,V and Sα,X admit another foliation by geodesics of the third type of Proposition 3.


### 4.1. Surfaces inL3 Foliated by Surface Constrained Critical Geodesics of Type (2)

As it is also customary, for a surfaceMr2 in Minkowski space we require the first fundamental form to be nondegenerate. This means, in particular, that our surfaces can not be compact. A surface is calledspacelike orRiemannian, M2 (r=0), if the first fundamental form is positive definite; it is calledtimelike orLorentzian, M12, if the first fundamental form is indefinite. In this section, for a given surface x:Mr2→L3, r∈{0,1}, we denote by ∇~ the Levi-Civita connection on L3 associated with the metric , (15) and by ∇ the Levi-Civita connection of the immersion (Mr2,x). Let h be the second fundamental form of Mr2 in L3. From now on, X, Y, Z, and W represent vector fields tangent to Mr2 and ξ denotes a vector field normal to Mr2. Then, the formulas of Gauss and Weingarten are [13](19)∇~XY=∇XY+hX,Y,(20)∇~Xξ=-AξX+DX⊥ξ.Here, D⊥ denotes the connection on the normal bundle of Mr2.If we denote the Riemann curvature tensor of∇ by R, then the equations of Gauss and Codazzi are given, respectively, by [13](21)RX,YZ,W=hX,W,hY,Z-hX,Z,hY,W,(22)∇hX,Y,Z=∇hY,X,Z,where(23)∇hX,Y,Z=DX⊥hY,Z-h∇XY,Z-hY,∇XZ.Now choosing an adapted local orthonormal frame e1,e2,e3 in L3 such that the vectors e1, e2 are tangent to Mr2 and e3 is normal to Mr2 in L3 and denoting by ω1,ω2,ω3 the dual frame of {e1,e2,e3}, theCartan connection forms are defined by(24)∇~Xei=∑jɛ~jωijXej,for i,j∈{1,2,3}, where ɛ~j=ej,ej is the causal character of ej. Then, ωij=-ωji and(25)hei,ej=ε~3hije3,hij=-∇~eie3,ej=ωj3ei,i,j∈{1,2} [13].Letx:Mr2→L3 denote a surface in Lorentzian 3-space L3 (if r=0, we simply denote Mr2=M2) with local orientation determined by the normal vector ξ. Take p∈Mr2 and let α:I→Mr2 be an immersed nonnull curve α(t); that is, dα/dt is a nonnull vector for all t∈I with causal character ε4, such that α(I) is contained in a local chart around p and α(0)=p. For any to∈I, take vo as a unit vector tangent to Mr2 at α(to) so that {dα/dt(to),vo,ξ(to)} form an orthonormal basis and consider the geodesic βto(s) with initial data: βto(0)=α(to) and dβto/ds(0)=vo. For the local parametrization of Mr2 in a neighbourhood of p defined by(26)xs,t=βts,t∈I, the coefficients of the metric are (reparametrizing the geodesics if needed) g11=xs,xs=-1rε4, g12=g21=xs,xt=0, and g22=xt,xt, which, for simplicity, is denoted by g22=ε4G2. That is, with respect to this parametrization, the metric can be written as(27)g=-1rε4ds2+ε4G2dt2.Using the metric coefficientsgij, one may compute the Christoffel symbols of the Levi-Civita connection of (27) with respect to this parametrization (see, e.g., [13], Proposition  1.1). In our case, we have(28)Γ111=Γ112=Γ121=0,Γ122=GsG,Γ221=-1r+1GGs,Γ222=GtG,where subscripts s and t mean partial derivative with respect to s and t, respectively. Hence the Levi-Civita connection of Mr2 is given by(29)∇∂/∂s∂∂s=0,∇∂/∂s∂∂t=GsG∂∂t,∇∂/∂t∂∂t=-1r+1GGs∂∂s+GtG∂∂t.Moreover, combining these equations and (21) we obtain that the Gaussian curvature of Mr2 is given by(30)K≔R∂/∂s,∂/∂t,∂/∂t,∂/∂sg∂/∂s,∂/∂s∂/∂t,∂/∂t-g∂/∂s,∂/∂t2=-1r+1ε4GssG.Ifβts, ∀s, were also a geodesic in L3, then Mr2 would be a ruled surface. So assume that βts is not a geodesic in L3, then ∇~β˙tβ˙t=β¨ts is not null (upper β˙ meaning derivative with respect to s) and the unit Frenet normal to βts is parallel to the unit normal to Mr2, for s lying in a certain interval It. Let us denote by Ts,t,Ns,t,Bs,t the Frenet frame of βts as described in (16) and choose the following local adapted frame on Mr2:(31)e1=xs=T,(32)e2=xtG=B,(33)e3=ξ=-N,where ξ is the unit normal to Mr2. Then, combining (19), (20), (24), (25), and (16), one gets(34)ω12e1=0,ω12e2=GsG,ω13e1=h11=-κ,(35)ω13e2=ω23e1=h12=τ,ω23e2≔h22,where κ(s,t) and τ(s,t) denote the curvature and torsion of the curves βts.The second fundamental form can be considered as a quadratic form given byhX≔AξX,X; therefore, we obtain from (35) that(36)-1rh=-κds2+2τGdsdt+G2h22dt2,with respect to the parametrization (26). Now, combining Gauss and Codazzi equations (21), (22) with (24), (29), (31), and (35), we obtain after a long computation(37)-1r+1ε4GssG=κh22+τ2.(38)κt=-2Gsτ-τsG,(39)τt=-1rκGs+h22Gs.Using Gauss formula (19) and the expression for Levi-Civita connection (29), one sees that the immersion (Mr2,x) satisfies the following PDE system:(40)xss=-1rκe3=-κGxs×xt,(41)xst=GsGxt-τGκxss,(42)xtt=-1r+1GsGxs+GtGxt-G2h22κxss.Conversely, observe that (37)–(39) are the compatibility conditions for the PDE system (40)–(42). Thus, given functions k(s,t), τ(s,t), G(s,t), and h22(s,t) smoothly defined on a connected domain U and satisfying (37)–(39), there exists a solution of (40)–(42) determining a smooth immersion (U,x) of a surface in L3 whose metric and second fundamental form are given by (27) and (36), respectively. This surface is foliated by geodesics having k(s,t) and τ(s,t) as curvature and torsion. If, in addition, U is simply connected, then the immersion (U,x) would be unique (up to rigid motions in L3). The regularity assumptions made before on the functions k(s,t), τ(s,t), G(s,t), and h22(s,t) can be significantly relaxed in several ways. For instance, if they were of class C2(U), then we would have an immersion x which would be C2(U). For weaker regularity conditions, one may see [19, 20], but here we restrict ourselves to the smooth case.A nonnull unit speed curve ofL3 with τ=0 lies in an affine plane. From now on, a curve with τ=0 is going to be called aplanar curve. We want to prove the following result.Proposition 4.
Letδ:I1→L3 be a nonnull arc-length parametrized curve δ(t) in L3, and let {Tδ(t),Nδ(t),Bδ(t)} denote either its Frenet frame given in (16) if δ′′(t) is nonnull or the frame given in (18) if δ′′(t) is null. We also denote by Pto≔span{Nδ(to),Bδ(to)} the normal plane to δ(t) at to∈I1. A
Suppose first thatδ(t) is spacelike and take any nonnull arc-length parametrized curve γ(s)=γ1(s),γ2(s) in the timelike plane Pto, γ:I2→Pto. Then, one has the following.
1
Ifδ′′(t) is nonnull, define the surface x:U=I1×I2→L3 given by(43)xs,t=δt+γ1scosh⁡ςtNδt+sinh⁡ςtBδt+γ2scosh⁡ςtBδt+sinh⁡ςtNδt,whereς satisfies ϵς′(t)=τδ(t), τδ(t) denotes the torsion of δ(t), and ϵ is the causal character of δ′′(t). Then, the immersion (U,xa) given in (43) defines a surface of L3 foliated by planar geodesics.(2)
Ifδ′′(t) is null, consider the surface x:U=I1×I2→L3 defined by(44)xas,t=aδt+γ1s12dcosh⁡ςt-sinh⁡ςtNδt+dcosh⁡ςt+sinh⁡ςtBδt+γ2s12dsinh⁡ςt-cosh⁡ςtNδt+dcosh⁡ςt+sinh⁡ςtBδt,whered∈R, a∈{0,1}, and ς′(t)=τδ(t) is the torsion of δ(t). Then, the immersion (U,xa) given in (44) defines a surface of L3 foliated by planar geodesics of (U,xa).Moreover, in both cases, the pseudo-Riemannian character of the surface is determined by that ofγ; that is, (U,xa) is Riemannian (resp., Lorentzian) if and only if γ is spacelike (resp., timelike).B
Assume now thatδ(t) is timelike. Take any nonnull arc-length parametrized curve γ(s)=γ1(s),γ2(s) in the spacelike plane Pto. Then,(45)xas,t=aδt+γ1scos⁡ςtNδt-sin⁡ςtBδt+γ2scos⁡ςtBδt+sin⁡ςtNδt,wherea∈{0,1} and ς′(t)=τδ(t) is the torsion of δ(t). Then, the immersion (U,xa) given in (45) defines a Lorentzian surface of L3 foliated by planar geodesics of (U,xa).
Conversely, locally, any surfaceMr2 in L3 foliated by nonnull planar geodesics is either a ruled surface or it can be constructed as described in (43), (44), and (45).Proof.
That the family of coordinate curvesxa(s,·) defined in (43), (44), and (45) gives a foliation of (U,xa) by planar geodesics can be checked by direct computation. Observe that the planar geodesics of the foliation are lines of curvature of the surface which are all congruent to γ. Their principal curvature is ɛκ(s), where ɛ denotes the causal character of δ(t) and κ(s) is the curvature of γ.
For the converse, assume thatMr2 is a surface in L3 foliated by planar geodesics which is not a ruled surface. Consider a curve α cutting orthogonally to the planar geodesic foliation and parametrize Mr2 locally as in (26). Take p∈Mr2 and let α:I→Mr2 be an immersed Frenet curve α(t) such that α(I) is contained in a local chart around p and α(0)=p and is perpendicular to the planar geodesic foliation. Since τ=0, then (37)–(39) reduce to(46)κt=0,(47)0=-1rκGs+h22Gs,(48)-1r+1ε4GssG=-κh22and (40), (41), and (42) to(49)xss=-1rκe3=-κGxs×xt,(50)xst=GsGxt,(51)xtt=-1r+1GsGxs+GtGxt-G2h22κxss.
Since our curvesβt(s), s∈Is, are not geodesics in the Lorentz space κ≠0, from (46) we have that κ(s) depends only on s. From now on, differentiation of one-variable functions ϕ(s),φ(t) will be denoted by ϕ˙(s)=dϕ(s)/ds and φ′(t)=dφ(t)/dt, respectively.
Case  1 (Riemannian surfaces:
r
=
0
,
ε
4
=
1
). Since the surface is Riemannian satisfying (27), we must have 〈xs,xs〉=1, 〈xs,xt〉=0, 〈xt,xt〉=G2, and κ/Gxs×xt,κ/Gxs×xt=-κ2. From (50) we have xt=Gs,tAt, for some At∈L3, t∈I1, which combined with xt2=G2 gives(52)xt=Gs,tTδt=Gs,tBs,t,where B(s,t)=Tδ(t)=δ′(t), for some unit speed spacelike curve δ(t) in L3; that is, B(s,t), the binormal of βt(s), is constant along the curve, as we already know since τ=0.
Ifδ(t) were a geodesic in L3, then δ′=V would be a unit constant vector and 〈xs,xt〉=0 would imply that xs lies in a plane perpendicular to V, πV. Hence, xst∈πV and combining (49), (50), (51), and (52), we obtain Gs=0 which means that x(s,t)=f(t)V+α(s), with α(s) being a unit speed curve in πV and f′(t)=G(t). Thus, our surface would be a right circular cylinder, Cα, shaped on a planar spacelike curve α contained in πV. The causal character of Cα is determined by that of α. So we may assume δ(t) is not a geodesic. Now, we distinguish two cases.
Case  1.1 (
r
=
0
and
δ
is a Frenet curve:
δ
′
′
is not a null vector). From 〈xs,xt〉=0 and (52), we have(53)xs=b1s,tNδt+b2s,tBδt,where {Tδ(t),Nδ(t),Bδ(t)} represents the Frenet frame along δ(t). Denote by ϵ1δ, ϵ2δ the causal characters of Nδ(t) and Bδ(t), respectively. Since δ(t) is spacelike, we must have ϵ1δϵ2δ=-1.
Assume firstϵ1δ=1 and ϵ2δ=-1; that is, δ′′ is spacelike. Then, xs,xt=0 implies(54)xs=cosh⁡θs,tNδt+sinh⁡θs,tBδt,for some function θ(s,t). Now, combining this equation with (50), (52), and (53) and using the Frenet equations for δ(t), (16), one has(55)Gs=-κδtcosh⁡θs,t,τδt=θts,t,where κδ(t), τδ(t) stand for the curvature and torsion of δ(t). Hence, θ(s,t)=ς1(s)+ς2(t), with τδ(t)=θt(s,t)=ς2′(t). Moreover, from (17), (49), and (54), we have(56)κs=-ς˙1s=-θss,t. Then, from the second equation of (55), we get(57)ηs,t=θs,t=ς1s+ς2t,ς2′t=τδt.On the other hand, a combination of (54) and (57) results in(58)xss,t=cosh⁡ς1scosh⁡ς2tNδt+sinh⁡ς2tBδt+sinh⁡ς1ssinh⁡ς2tNδt+cosh⁡ς2tBδt.Define(59)γs≔γ1sV1+γ2sV2,γ1s≔∫soscosh⁡ς1vdv,γ2s≔∫sossinh⁡ς1vdv,V1≔cosh⁡ς2toNδto+sinh⁡ς2toBδto,V2≔cosh⁡ς2toBδto-sinh⁡ς2toNδto,so that γ(s)=γ1(s),γ2(s) is the only (up to isometries) spacelike curve contained in the timelike plane Pto≔spanV1,V2 with curvature κ(s). Then, (58) gives(60)xs,t=Dt+γ1scosh⁡ς2tNδt+sinh⁡ς2tBδt+γ2scosh⁡ς2tBδt+sinh⁡ς2tNδt,where D(t) is a curve in L3. Finally, using (60) along with (52), (55), and (57) we obtain after some computations(61)D′t=μtδ′t,where μ(t)=-κδ(t)λ1cosh⁡ς2(t)+λ2sinh⁡ς2(t), λi∈R. Observe that μ(t)≠0 since δ(t) is not a geodesic. Hence, from (61) we have D′′t,D′′t=μ′(t)2+μ(t)2>0, since δ′, δ′′ are spacelike vectors. Thus, if we reparametrize D(t) by the arc-length parameter u and calling the reparametrized curve mu=Dtu, one can check that d2m/du2 is nonnull and(62)Nmu=Nδtu,Bmu=Bδtu,τmu=τδtuμtu,where the subscript m is meant to denote the geometric elements associated with m(u). Hence, (60) can be written as(63)ys,u≔xs,tu=mu+γ1scosh⁡ς^2uNmu+sinh⁡ς^2uBmu+γ2scosh⁡ς^2uBmu+sinh⁡ς^2uNmu,where mu is a spacelike curve in L3, {Nm,Bm} are the normal and binormal unit vectors of the Frenet frame along m(u), ς^2(u)≔ς2(t(u)),d/duς^2(u)=τm(u) is the torsion of m(u), and γ(s) is the unique spacelike curve with curvature κs (up to isometries) contained in the timelike plane Puo=u-1(to) described before. So this case falls under (A)(1).
If now we assume thatϵ1δ=-1 and ϵ2δ=1 (i.e., δ′′ is timelike), then analogous arguments lead to(64)θs,t=ς1s+ς2t,ς2′t=-τδt,κs=ς˙1s,xss,t=sinh⁡ς1scosh⁡ς2tNδt+sinh⁡ς2tBδt+cosh⁡ς1ssinh⁡ς2tNδt+cosh⁡ς2tBδt.So we arrive to similar conclusions by defining γ2(s)≔∫soscosh⁡ς1(v)dv, γ1s≔∫sossinh⁡ς1(v)dv. Thus, this case is also covered in (A)(1).
Case  1.2 (
r
=
0
and
δ
′
′
is a null vector). From xs,xt=0 and (52), we have(65)xs=b1s,tNδt+b2s,tBδt,where {Tδ(t),Nδ(t),Bδ(t)} represents the frame along δ(t) defined in (18). Then, 〈xs,xs〉=1 implies(66)2b1s,tb2s,t=1.Now, combining this equation with (52) and (65) and using the Frenet equations for δ(t), (18), one has(67)Gs=-b2s,t,b1′s,t+b1s,tτδt=0,b2′s,t-b2s,tτδt=0.With integration, we obtain(68)Gs=-b2s,t=μ2se∫τδtdt,b1s,t=μ1se-∫τδtdt,μ1sμ2s=-12.Then, combining (47) and (48) one has(69)Gs2-Gh222=±a2t,or equivalently(70)Gs2-Gssκ2=±a2t.Hence, from (70) and (68) one has(71)a2t=e2∫τδtdtμ22s-μ˙22sκ2s.Therefore,(72)μ22s-μ˙22sκ2s=λ,for a certain constant λ∈R. Observe that μ˙22s≠0; otherwise b1(t) and b2(t) would depend only on t and xss would be zero. This would contradict (49) since we are assuming κ(s)≠0. Thus, by differentiating (65) and using (68), (71), and xss,xss=-κ2s, one gets that a2(t)=λ=0 in this case. Then, solving (72) we have(73)μ2s=-de±∫κsds,so (68) gives(74)Gs,t=de∫τδtdt∫e±∫κsds+η2t,with d∈R. Combining (65), (67), and (68), we obtain(75)xs,t=12de-∫τδtdt∫e∓∫κsdsdsNδt+de∫τδtdt∫e±∫κsdsdsBδt+Dt,where D(t) is a curve in L3. Then, (52) gives(76)D′t=η2tTδt.Assume first that μ2(s)=-de∫κ(s)ds in (73). Now, defining ς1(s) by ς˙1(s)=κ(s), consider the planar curve γ(s)≔γ1(s)V1+γ2(s)V2, in the timelike plane Pto≔span{Nδ(to),Bδ(to)} with coordinates γ1(s)≔∫sos‍cosh⁡ς1(v)dv and γ2(s)≔∫sos‍sinh⁡ς1(v)dv with respect to the frames V1≔Nδ(to)+1/2Bδ(to) and V2≔Nδ(to)-1/2Bδ(to). That is, γ(s)=γ1(s),γ2(s) is the only (up to isometries) spacelike curve contained in the timelike plane Pto≔span{V1,V2} with curvature κ(s). Then, (75) can be written in the following way:(77)xs,t=γ1s12dcosh⁡ςt-sinh⁡ςtNδt+dcosh⁡ςt+sinh⁡ςtBδt±γ2s12dsinh⁡ςt-cosh⁡ςtNδt+dcosh⁡ςt+sinh⁡ςtBδt+Dt,with D(t) being a curve in L3 and ς′(t)=τδ(t).
From (76), we have D′t,D′t=η22t≥0. If η2(t)=0, then D(t) is constant. Thus, we assume η2(t)≠0. From (76), we have D′′t,D′′t=η2′(t)2≥0. If η2′(t)=0, η2(t) is constant and D(t) is a reparametrization of δ(t) at constant speed. If η2′(t)≠0, we reparametrize D(t) by the arc-length parameter u and call m(u)=Dtu to the reparametrized curve. In this case, one can check that d2m/du2 is null and(78)Nmu=1η2tuNδtu,Bmu=η2tuBδtu,τmu=-1η2tuη2tudu+τδtuη2tu.Then, (77) can be written as(79)ys,u≔xs,tu=mu+γ1s12dcosh⁡ς^u-sinh⁡ς^uNmu+dcosh⁡ς^u+sinh⁡ς^uBmu+γ2s12dsinh⁡ς^u-cosh⁡ς^uNmu+dcosh⁡ς^u+sinh⁡ς^uBmu,where mu is a spacelike curve in L3, {Nm,Bm} are the normal and binormal unit vectors of the Frenet frame along m(u), ς^(u)≔ς(t(u)), d/duς^(u)=τm(u) (torsion of m(u)), and γ(s) is the unique spacelike curve with curvature κ(s) (up to isometries) contained in the timelike plane Puo=u-1(to) described before. So this case falls under (A)(2).
Ifμ2(s)=-de-∫‍κ(s)ds in (73), then, taking now ς˙1(s)=-κ(s), one can repeat the argument following (76) and gets a similar conclusion. This concludes Case  1.
Case  2 (Lorentzian surface
r
=
1
). If r=1, we have that x:M12→L3 is a Lorentzian surface. For the sake of brevity, most of computations in this case will be omitted. Since the geodesics of the foliation are not null, we distinguish two cases according to the causal character of α.
Case  2.1 (
α
is spacelike). With respect to the local parametrization of M12 in a neighbourhood of p defined by(80)xs,t=βts,t∈I, the metric can be written as(81)g=-ds2+G2dt2.Now, we have r=1 and ε4=1 in the PDE system (49), (50), and (51), the second equation of which gives again(82)xt=Gs,tTδt=Gs,tBs,t.Defining δ(t) by δ′(t)=Tδ(t), one has that δ(t) is spacelike and, using again similar arguments to those applied in the previous case (Riemannian surface, r=0), one can verify after a long computation that the same two cases (A)(1) and (A)(2) of Proposition 4 are obtained also in this case.
Case  2.2 (
α
is timelike). Let us suppose now that α(t) is a timelike curve. Then, the expression of the metric in the local parametrization we are using is(83)g=ds2-G2dt2,and we have to use r=1 and ε4=-1 in the PDE system (49), (50), and (51), again the second equation of which gives(84)xt=Gs,tTδt=Gs,tBs,t,where δ(t) is such that δ′t=Tδ(t)=Bs,t is a unit timelike vector field. Therefore, since δ′t=Bs,t, δ′′ falls into the spacelike plane generated by T(s,t)=xs and ξ, because δ′,δ′′=0 and δ′′ is not null. Now,(85)xs=cos⁡θs,tNδ+sin⁡θs,tBδ,and one can use again similar arguments to those applied in case of a Riemannian surface for nonnull δ′′, substituting cosh and sinh by cos and sin, respectively, obtaining case (B) of Proposition 4.


### 4.2. Hashimoto Surfaces Foliated by Constrained Critical Geodesics of Type (3)

In 1906, da Rios [21] modeled the movement of a thin vortex filament in a viscous fluid by the motion of a curve propagating in R3 according to(86)∂x∂t=∂x∂s×∂2x∂s2,which is known as thelocalized induction equation, LIE (see [22]). For notation consistency, LIE is often to be written as(87)xt=xs×xss.In this section we are going to consider the evolution inL3 of a Frenet curve, γ(s), of rank 2 or 3 under LIE (87). Let x(s,t) describe the evolution of γ(s) under LIE and denote x(s,t)=γt(s), γ0(s)=γ(s), s∈(-ɛ,ɛ), and t∈(-δ,δ), where t represents the time evolution parameter. It is easy to show that if s is the proper time for γ(s), that is, the arc-length parameter, then so is the case for every t∈(-δ,δ). In fact, using (87) we have(88)∂∂tγ˙ts,γ˙ts=2det⁡∂x∂s,∂3x∂s3,∂x∂s=0;that is, 〈∂γt/∂s,∂γt/∂s〉 does not depend on “time” t, so since 〈∂γ0/∂s,∂γ0/∂s〉=〈∂γ/∂s,∂γ/∂s〉=〈T,T〉=ε1, then so is the case for every t∈(-δ,δ).From now on, we will assume thats is the arc-length parameter and that xss is nonnull everywhere. Then for any fix to we may consider the associated Frenet frame {T=γ˙,N,B}(s,to) on x(s,to)=γto(s) described in (16). We are going to assume also that x(s,t) defines an immersed surface in L3 which will be called aHashimoto surface (with initial condition γ) Sγ. For any given to, the curve x(s,to) will be referred to as avortex curve. If every vortex curve is a C4 closed curve, the Hashimoto surface will be calledHashimoto tube.Since our curvesγ are arc-length parametrized, LIE can be simplified in terms of thebinormal flow. To be more precise, using (16) and (17) in (87) we have(89)∂x∂t=∂x∂s×∂2x∂s2=T×DTds=T×ε2κN=ε2ε3κB.This means that if γ is a Frenet curve of rank 2 or 3 parametrized by arc-length and evolving under LIE (87) with x(s,0)=γ0(s)=γ(s), then it evolves by the binormal flow.Observe that for a Hashimoto surfaceSγ the filament evolution x(s,t) under LIE implies that the vortex curves (t-curves) x(s,to) are geodesics in Sγ and then x(s,t) gives a parametrization of Sγ of type (26) where, as a consequence of (89), the induced metric is expressed as in (27) with G=ε2ε3κ, with κ being the curvature of an orthogonal curve to the geodesic foliation of Sγ determined by x(s,t). Hence, one can see that the Gauss-Codazzi equations (37)–(39) and the PDE system (40)–(42) reduce, respectively, to(90)κt=-ε2ε32κsτ+κτs,(91)τt=ε2ε2κssκ-ε3τ2+12ε1κ2s,(92)xss=ε2κN,(93)xst=κsκxt-ε3κτN,(94)xtt=-ε1ε3κκsxs+ε2ε3-2κsτκ-τsxt+ε2κτ2-ε3κssN.Notice that if we were considering evolution under LIE in the standard Euclidean case, then εi=1, for i=1,2,3, and (90) and (91) would be the well-knownda Rios equations [21]. In other words, in the Riemannian case da Rios equations are nothing but the Gauss-Codazzi equation of Hashimoto surfaces expressed with respect to the geodesic coordinate system (26). By this reason, the Gauss-Codazzi equations of the Lorentzian case, (90) and (91), will be referred to as theLorentzian da Rios equations.Lorentzian Hashimoto surfaces have the following properties.Proposition 5.
With the previous notation, letSγ be a Hashimoto surface having by initial condition a Frenet curve of rank 2 or 3 in L3, γ(s), parametrized by proper time. Denote by x(s,t) the parametrization of Sγ determined by LIE (87). Then, one has the following. (1)
Ifs∈[0,1] and all vortex curves are C4-closed in [0,1], then ∫o1κ2ds and ∫o1τds are independent of t.(2)
If all vortex curves are planar, then they are elasticae in either a Riemannian or a Lorentzian plane. The corresponding Hashimoto surface is described in Proposition4 and if δ′′ is not null, then Sγ is either a right cylinder on a Lorentzian circle or a rotation surface shaped on a planar elastica γ of either R2 or L2.(3)
The initial vortex curveγ(s) evolves by rigid motions under LIE, if and only if it is an elastica in L3. As a consequence, a rank 3 elastica γ(s) in L3 evolves under LIE (by rigid motions) and the different positions of the vortex curve over time give foliation of the associated Hashimoto surface by Sγ-constrained elastic geodesics of type 3 in Proposition 3.Proof.
Assume thats∈[0,1] and that, for every t, x(s,t) is C4-closed in [0,1]. Then, using (90),(95)ddt∫01κ2s,tds=2∫01κtκds=-2ε2ε3∫01ddsκ2τds=0.Thus, ∫o1‍κ2ds attains the same value at every vortex curve. That ∫o1‍τds does not depend on time evolution follows trivially from (91).
Now, we want to prove2. If all vortex curves are planar, then τ=0 and, therefore, since they are not null, they must lie either in an Euclidean plane R2 or in a Lorentzian plane L2. Moreover, (90) implies that κ(s,t)=κ(s) does not depend on t and (91) gives ε2κss+1/2ε1κ3+μκ=0, μ∈R, which is precisely the equation for an elastica either in R2 (ε1=ε2=1) or in L2 (ε1ε2=-1) [6]. On the other hand, since τ=0, we have that B(s,t)=B(t) does not depend on s. If κ were constant, then vortex curves are circles (planar curves with constant curvature in L3) and, after differentiation of (89) with respect to t combined with the Frenet equations (16), we get that B(t) is a constant vector of L3. Hence, Sγ is a right circular cylinder. If κ is not constant, we combine (93) with (89) to obtain(96)κκδNδ=-ε1ε3κκsxs-ε2ε3κssκxss,where we are using the notation of Proposition 4 and assuming that δ′′ is not null. Differentiating (96) with respect to t and using (16) again, we obtain that κδ is constant and τδ=0. Hence, δ(t) is a planar circle. Assume first that δ is timelike. Then, Bδ is a spacelike constant vector that we may choose to be (0,0,1). Thus, δ can be parametrized as δ(s)=(1/kosinh⁡kot,1/kocosh⁡kot,0). Hence, by repeating the argument in the proof of Proposition 4, we have(97)xs,t=κskosinh⁡kot,κskocosh⁡kot,as,where κs2/ko2+as2=1 and the curve γ(s)=(κ(s),a(s)) is an elastica either in R2 or in L2. Thus Sγ is a rotation surface with profile curve γ. A similar conclusion can be achieved if δ is spacelike. This finishes item 2.
Finally, let us prove3. If γ evolves by rigid motion of L3, then κ(s,t)=κ(s) and τ(s,t)=τ(s) do not depend on t, so the left hand sides of (90) and (91) are zero. In particular, we get (98)0=2κsτ+κτs,0=2ε2κss-2ε3τ2κ+ε1κ3+μκ,μ∈R, which are the equations for elasticae in L3 [6]. In particular, γ is an elastica in L3.
Conversely, assume thatγ(s)=x(s,0) is an elastica in γ(s); then, the following vector field along γ(s),(99)Js=ε2ε3κs,0Bs,0,can be extended to a Killing field on L3, V [6]. Denote by {ϕt,t∈R} the 1-parameter group of isometries of L3 associated with V and consider y(s,t)≔ϕt(γ(s)). Then, since ϕt are isometries, we have γ(s)=y(s,0) and ε2ε3κs,tBs,t=yt(s,t), what implies that y(s,t)=x(s,t) and x(s,t) describes an evolution of γ(s)=x(s,0) under LIE by rigid motions.
As for the last part of statement3, if γ(s) is a rank 3 elastica in L3, then we already know that it evolves under LIE by rigid motions. Therefore, the different positions of the vortex curve over time, x(s,t), are also rank 3 elastica in L3. Trivially, elasticae in L3 are also Sγ-constrained elasticae; thus, x(s,t) gives a foliation of the associated Hashimoto surface, Sγ, by Sγ-constrained elastic geodesics of type 3 in Proposition 3.Remark 6.
Observe that these facts are an extension of the corresponding properties in the Riemannian setting. In fact, takingεi=1, i∈1,2,3, in the above proof, we see that these properties are also true for Hashimoto surfaces in R3. In this case, a somehow different proof of items (2) and (3) of Proposition 5 can be found in [23] and [24], respectively.
Also, in connection with item2 of Proposition 5, it should be mentioned that as it has been proved in [25], Hashimoto surfaces of revolution in R3 (resp., in L3) must be shaped on elasticae of R2 (resp., of L2) and, moreover, they provide congruence solutions of (87).


## 4.1. Surfaces inL3 Foliated by Surface Constrained Critical Geodesics of Type (2)

As it is also customary, for a surfaceMr2 in Minkowski space we require the first fundamental form to be nondegenerate. This means, in particular, that our surfaces can not be compact. A surface is calledspacelike orRiemannian, M2 (r=0), if the first fundamental form is positive definite; it is calledtimelike orLorentzian, M12, if the first fundamental form is indefinite. In this section, for a given surface x:Mr2→L3, r∈{0,1}, we denote by ∇~ the Levi-Civita connection on L3 associated with the metric , (15) and by ∇ the Levi-Civita connection of the immersion (Mr2,x). Let h be the second fundamental form of Mr2 in L3. From now on, X, Y, Z, and W represent vector fields tangent to Mr2 and ξ denotes a vector field normal to Mr2. Then, the formulas of Gauss and Weingarten are [13](19)∇~XY=∇XY+hX,Y,(20)∇~Xξ=-AξX+DX⊥ξ.Here, D⊥ denotes the connection on the normal bundle of Mr2.If we denote the Riemann curvature tensor of∇ by R, then the equations of Gauss and Codazzi are given, respectively, by [13](21)RX,YZ,W=hX,W,hY,Z-hX,Z,hY,W,(22)∇hX,Y,Z=∇hY,X,Z,where(23)∇hX,Y,Z=DX⊥hY,Z-h∇XY,Z-hY,∇XZ.Now choosing an adapted local orthonormal frame e1,e2,e3 in L3 such that the vectors e1, e2 are tangent to Mr2 and e3 is normal to Mr2 in L3 and denoting by ω1,ω2,ω3 the dual frame of {e1,e2,e3}, theCartan connection forms are defined by(24)∇~Xei=∑jɛ~jωijXej,for i,j∈{1,2,3}, where ɛ~j=ej,ej is the causal character of ej. Then, ωij=-ωji and(25)hei,ej=ε~3hije3,hij=-∇~eie3,ej=ωj3ei,i,j∈{1,2} [13].Letx:Mr2→L3 denote a surface in Lorentzian 3-space L3 (if r=0, we simply denote Mr2=M2) with local orientation determined by the normal vector ξ. Take p∈Mr2 and let α:I→Mr2 be an immersed nonnull curve α(t); that is, dα/dt is a nonnull vector for all t∈I with causal character ε4, such that α(I) is contained in a local chart around p and α(0)=p. For any to∈I, take vo as a unit vector tangent to Mr2 at α(to) so that {dα/dt(to),vo,ξ(to)} form an orthonormal basis and consider the geodesic βto(s) with initial data: βto(0)=α(to) and dβto/ds(0)=vo. For the local parametrization of Mr2 in a neighbourhood of p defined by(26)xs,t=βts,t∈I, the coefficients of the metric are (reparametrizing the geodesics if needed) g11=xs,xs=-1rε4, g12=g21=xs,xt=0, and g22=xt,xt, which, for simplicity, is denoted by g22=ε4G2. That is, with respect to this parametrization, the metric can be written as(27)g=-1rε4ds2+ε4G2dt2.Using the metric coefficientsgij, one may compute the Christoffel symbols of the Levi-Civita connection of (27) with respect to this parametrization (see, e.g., [13], Proposition  1.1). In our case, we have(28)Γ111=Γ112=Γ121=0,Γ122=GsG,Γ221=-1r+1GGs,Γ222=GtG,where subscripts s and t mean partial derivative with respect to s and t, respectively. Hence the Levi-Civita connection of Mr2 is given by(29)∇∂/∂s∂∂s=0,∇∂/∂s∂∂t=GsG∂∂t,∇∂/∂t∂∂t=-1r+1GGs∂∂s+GtG∂∂t.Moreover, combining these equations and (21) we obtain that the Gaussian curvature of Mr2 is given by(30)K≔R∂/∂s,∂/∂t,∂/∂t,∂/∂sg∂/∂s,∂/∂s∂/∂t,∂/∂t-g∂/∂s,∂/∂t2=-1r+1ε4GssG.Ifβts, ∀s, were also a geodesic in L3, then Mr2 would be a ruled surface. So assume that βts is not a geodesic in L3, then ∇~β˙tβ˙t=β¨ts is not null (upper β˙ meaning derivative with respect to s) and the unit Frenet normal to βts is parallel to the unit normal to Mr2, for s lying in a certain interval It. Let us denote by Ts,t,Ns,t,Bs,t the Frenet frame of βts as described in (16) and choose the following local adapted frame on Mr2:(31)e1=xs=T,(32)e2=xtG=B,(33)e3=ξ=-N,where ξ is the unit normal to Mr2. Then, combining (19), (20), (24), (25), and (16), one gets(34)ω12e1=0,ω12e2=GsG,ω13e1=h11=-κ,(35)ω13e2=ω23e1=h12=τ,ω23e2≔h22,where κ(s,t) and τ(s,t) denote the curvature and torsion of the curves βts.The second fundamental form can be considered as a quadratic form given byhX≔AξX,X; therefore, we obtain from (35) that(36)-1rh=-κds2+2τGdsdt+G2h22dt2,with respect to the parametrization (26). Now, combining Gauss and Codazzi equations (21), (22) with (24), (29), (31), and (35), we obtain after a long computation(37)-1r+1ε4GssG=κh22+τ2.(38)κt=-2Gsτ-τsG,(39)τt=-1rκGs+h22Gs.Using Gauss formula (19) and the expression for Levi-Civita connection (29), one sees that the immersion (Mr2,x) satisfies the following PDE system:(40)xss=-1rκe3=-κGxs×xt,(41)xst=GsGxt-τGκxss,(42)xtt=-1r+1GsGxs+GtGxt-G2h22κxss.Conversely, observe that (37)–(39) are the compatibility conditions for the PDE system (40)–(42). Thus, given functions k(s,t), τ(s,t), G(s,t), and h22(s,t) smoothly defined on a connected domain U and satisfying (37)–(39), there exists a solution of (40)–(42) determining a smooth immersion (U,x) of a surface in L3 whose metric and second fundamental form are given by (27) and (36), respectively. This surface is foliated by geodesics having k(s,t) and τ(s,t) as curvature and torsion. If, in addition, U is simply connected, then the immersion (U,x) would be unique (up to rigid motions in L3). The regularity assumptions made before on the functions k(s,t), τ(s,t), G(s,t), and h22(s,t) can be significantly relaxed in several ways. For instance, if they were of class C2(U), then we would have an immersion x which would be C2(U). For weaker regularity conditions, one may see [19, 20], but here we restrict ourselves to the smooth case.A nonnull unit speed curve ofL3 with τ=0 lies in an affine plane. From now on, a curve with τ=0 is going to be called aplanar curve. We want to prove the following result.Proposition 4.
Letδ:I1→L3 be a nonnull arc-length parametrized curve δ(t) in L3, and let {Tδ(t),Nδ(t),Bδ(t)} denote either its Frenet frame given in (16) if δ′′(t) is nonnull or the frame given in (18) if δ′′(t) is null. We also denote by Pto≔span{Nδ(to),Bδ(to)} the normal plane to δ(t) at to∈I1. A
Suppose first thatδ(t) is spacelike and take any nonnull arc-length parametrized curve γ(s)=γ1(s),γ2(s) in the timelike plane Pto, γ:I2→Pto. Then, one has the following.
1
Ifδ′′(t) is nonnull, define the surface x:U=I1×I2→L3 given by(43)xs,t=δt+γ1scosh⁡ςtNδt+sinh⁡ςtBδt+γ2scosh⁡ςtBδt+sinh⁡ςtNδt,whereς satisfies ϵς′(t)=τδ(t), τδ(t) denotes the torsion of δ(t), and ϵ is the causal character of δ′′(t). Then, the immersion (U,xa) given in (43) defines a surface of L3 foliated by planar geodesics.(2)
Ifδ′′(t) is null, consider the surface x:U=I1×I2→L3 defined by(44)xas,t=aδt+γ1s12dcosh⁡ςt-sinh⁡ςtNδt+dcosh⁡ςt+sinh⁡ςtBδt+γ2s12dsinh⁡ςt-cosh⁡ςtNδt+dcosh⁡ςt+sinh⁡ςtBδt,whered∈R, a∈{0,1}, and ς′(t)=τδ(t) is the torsion of δ(t). Then, the immersion (U,xa) given in (44) defines a surface of L3 foliated by planar geodesics of (U,xa).Moreover, in both cases, the pseudo-Riemannian character of the surface is determined by that ofγ; that is, (U,xa) is Riemannian (resp., Lorentzian) if and only if γ is spacelike (resp., timelike).B
Assume now thatδ(t) is timelike. Take any nonnull arc-length parametrized curve γ(s)=γ1(s),γ2(s) in the spacelike plane Pto. Then,(45)xas,t=aδt+γ1scos⁡ςtNδt-sin⁡ςtBδt+γ2scos⁡ςtBδt+sin⁡ςtNδt,wherea∈{0,1} and ς′(t)=τδ(t) is the torsion of δ(t). Then, the immersion (U,xa) given in (45) defines a Lorentzian surface of L3 foliated by planar geodesics of (U,xa).
Conversely, locally, any surfaceMr2 in L3 foliated by nonnull planar geodesics is either a ruled surface or it can be constructed as described in (43), (44), and (45).Proof.
That the family of coordinate curvesxa(s,·) defined in (43), (44), and (45) gives a foliation of (U,xa) by planar geodesics can be checked by direct computation. Observe that the planar geodesics of the foliation are lines of curvature of the surface which are all congruent to γ. Their principal curvature is ɛκ(s), where ɛ denotes the causal character of δ(t) and κ(s) is the curvature of γ.
For the converse, assume thatMr2 is a surface in L3 foliated by planar geodesics which is not a ruled surface. Consider a curve α cutting orthogonally to the planar geodesic foliation and parametrize Mr2 locally as in (26). Take p∈Mr2 and let α:I→Mr2 be an immersed Frenet curve α(t) such that α(I) is contained in a local chart around p and α(0)=p and is perpendicular to the planar geodesic foliation. Since τ=0, then (37)–(39) reduce to(46)κt=0,(47)0=-1rκGs+h22Gs,(48)-1r+1ε4GssG=-κh22and (40), (41), and (42) to(49)xss=-1rκe3=-κGxs×xt,(50)xst=GsGxt,(51)xtt=-1r+1GsGxs+GtGxt-G2h22κxss.
Since our curvesβt(s), s∈Is, are not geodesics in the Lorentz space κ≠0, from (46) we have that κ(s) depends only on s. From now on, differentiation of one-variable functions ϕ(s),φ(t) will be denoted by ϕ˙(s)=dϕ(s)/ds and φ′(t)=dφ(t)/dt, respectively.
Case  1 (Riemannian surfaces:
r
=
0
,
ε
4
=
1
). Since the surface is Riemannian satisfying (27), we must have 〈xs,xs〉=1, 〈xs,xt〉=0, 〈xt,xt〉=G2, and κ/Gxs×xt,κ/Gxs×xt=-κ2. From (50) we have xt=Gs,tAt, for some At∈L3, t∈I1, which combined with xt2=G2 gives(52)xt=Gs,tTδt=Gs,tBs,t,where B(s,t)=Tδ(t)=δ′(t), for some unit speed spacelike curve δ(t) in L3; that is, B(s,t), the binormal of βt(s), is constant along the curve, as we already know since τ=0.
Ifδ(t) were a geodesic in L3, then δ′=V would be a unit constant vector and 〈xs,xt〉=0 would imply that xs lies in a plane perpendicular to V, πV. Hence, xst∈πV and combining (49), (50), (51), and (52), we obtain Gs=0 which means that x(s,t)=f(t)V+α(s), with α(s) being a unit speed curve in πV and f′(t)=G(t). Thus, our surface would be a right circular cylinder, Cα, shaped on a planar spacelike curve α contained in πV. The causal character of Cα is determined by that of α. So we may assume δ(t) is not a geodesic. Now, we distinguish two cases.
Case  1.1 (
r
=
0
and
δ
is a Frenet curve:
δ
′
′
is not a null vector). From 〈xs,xt〉=0 and (52), we have(53)xs=b1s,tNδt+b2s,tBδt,where {Tδ(t),Nδ(t),Bδ(t)} represents the Frenet frame along δ(t). Denote by ϵ1δ, ϵ2δ the causal characters of Nδ(t) and Bδ(t), respectively. Since δ(t) is spacelike, we must have ϵ1δϵ2δ=-1.
Assume firstϵ1δ=1 and ϵ2δ=-1; that is, δ′′ is spacelike. Then, xs,xt=0 implies(54)xs=cosh⁡θs,tNδt+sinh⁡θs,tBδt,for some function θ(s,t). Now, combining this equation with (50), (52), and (53) and using the Frenet equations for δ(t), (16), one has(55)Gs=-κδtcosh⁡θs,t,τδt=θts,t,where κδ(t), τδ(t) stand for the curvature and torsion of δ(t). Hence, θ(s,t)=ς1(s)+ς2(t), with τδ(t)=θt(s,t)=ς2′(t). Moreover, from (17), (49), and (54), we have(56)κs=-ς˙1s=-θss,t. Then, from the second equation of (55), we get(57)ηs,t=θs,t=ς1s+ς2t,ς2′t=τδt.On the other hand, a combination of (54) and (57) results in(58)xss,t=cosh⁡ς1scosh⁡ς2tNδt+sinh⁡ς2tBδt+sinh⁡ς1ssinh⁡ς2tNδt+cosh⁡ς2tBδt.Define(59)γs≔γ1sV1+γ2sV2,γ1s≔∫soscosh⁡ς1vdv,γ2s≔∫sossinh⁡ς1vdv,V1≔cosh⁡ς2toNδto+sinh⁡ς2toBδto,V2≔cosh⁡ς2toBδto-sinh⁡ς2toNδto,so that γ(s)=γ1(s),γ2(s) is the only (up to isometries) spacelike curve contained in the timelike plane Pto≔spanV1,V2 with curvature κ(s). Then, (58) gives(60)xs,t=Dt+γ1scosh⁡ς2tNδt+sinh⁡ς2tBδt+γ2scosh⁡ς2tBδt+sinh⁡ς2tNδt,where D(t) is a curve in L3. Finally, using (60) along with (52), (55), and (57) we obtain after some computations(61)D′t=μtδ′t,where μ(t)=-κδ(t)λ1cosh⁡ς2(t)+λ2sinh⁡ς2(t), λi∈R. Observe that μ(t)≠0 since δ(t) is not a geodesic. Hence, from (61) we have D′′t,D′′t=μ′(t)2+μ(t)2>0, since δ′, δ′′ are spacelike vectors. Thus, if we reparametrize D(t) by the arc-length parameter u and calling the reparametrized curve mu=Dtu, one can check that d2m/du2 is nonnull and(62)Nmu=Nδtu,Bmu=Bδtu,τmu=τδtuμtu,where the subscript m is meant to denote the geometric elements associated with m(u). Hence, (60) can be written as(63)ys,u≔xs,tu=mu+γ1scosh⁡ς^2uNmu+sinh⁡ς^2uBmu+γ2scosh⁡ς^2uBmu+sinh⁡ς^2uNmu,where mu is a spacelike curve in L3, {Nm,Bm} are the normal and binormal unit vectors of the Frenet frame along m(u), ς^2(u)≔ς2(t(u)),d/duς^2(u)=τm(u) is the torsion of m(u), and γ(s) is the unique spacelike curve with curvature κs (up to isometries) contained in the timelike plane Puo=u-1(to) described before. So this case falls under (A)(1).
If now we assume thatϵ1δ=-1 and ϵ2δ=1 (i.e., δ′′ is timelike), then analogous arguments lead to(64)θs,t=ς1s+ς2t,ς2′t=-τδt,κs=ς˙1s,xss,t=sinh⁡ς1scosh⁡ς2tNδt+sinh⁡ς2tBδt+cosh⁡ς1ssinh⁡ς2tNδt+cosh⁡ς2tBδt.So we arrive to similar conclusions by defining γ2(s)≔∫soscosh⁡ς1(v)dv, γ1s≔∫sossinh⁡ς1(v)dv. Thus, this case is also covered in (A)(1).
Case  1.2 (
r
=
0
and
δ
′
′
is a null vector). From xs,xt=0 and (52), we have(65)xs=b1s,tNδt+b2s,tBδt,where {Tδ(t),Nδ(t),Bδ(t)} represents the frame along δ(t) defined in (18). Then, 〈xs,xs〉=1 implies(66)2b1s,tb2s,t=1.Now, combining this equation with (52) and (65) and using the Frenet equations for δ(t), (18), one has(67)Gs=-b2s,t,b1′s,t+b1s,tτδt=0,b2′s,t-b2s,tτδt=0.With integration, we obtain(68)Gs=-b2s,t=μ2se∫τδtdt,b1s,t=μ1se-∫τδtdt,μ1sμ2s=-12.Then, combining (47) and (48) one has(69)Gs2-Gh222=±a2t,or equivalently(70)Gs2-Gssκ2=±a2t.Hence, from (70) and (68) one has(71)a2t=e2∫τδtdtμ22s-μ˙22sκ2s.Therefore,(72)μ22s-μ˙22sκ2s=λ,for a certain constant λ∈R. Observe that μ˙22s≠0; otherwise b1(t) and b2(t) would depend only on t and xss would be zero. This would contradict (49) since we are assuming κ(s)≠0. Thus, by differentiating (65) and using (68), (71), and xss,xss=-κ2s, one gets that a2(t)=λ=0 in this case. Then, solving (72) we have(73)μ2s=-de±∫κsds,so (68) gives(74)Gs,t=de∫τδtdt∫e±∫κsds+η2t,with d∈R. Combining (65), (67), and (68), we obtain(75)xs,t=12de-∫τδtdt∫e∓∫κsdsdsNδt+de∫τδtdt∫e±∫κsdsdsBδt+Dt,where D(t) is a curve in L3. Then, (52) gives(76)D′t=η2tTδt.Assume first that μ2(s)=-de∫κ(s)ds in (73). Now, defining ς1(s) by ς˙1(s)=κ(s), consider the planar curve γ(s)≔γ1(s)V1+γ2(s)V2, in the timelike plane Pto≔span{Nδ(to),Bδ(to)} with coordinates γ1(s)≔∫sos‍cosh⁡ς1(v)dv and γ2(s)≔∫sos‍sinh⁡ς1(v)dv with respect to the frames V1≔Nδ(to)+1/2Bδ(to) and V2≔Nδ(to)-1/2Bδ(to). That is, γ(s)=γ1(s),γ2(s) is the only (up to isometries) spacelike curve contained in the timelike plane Pto≔span{V1,V2} with curvature κ(s). Then, (75) can be written in the following way:(77)xs,t=γ1s12dcosh⁡ςt-sinh⁡ςtNδt+dcosh⁡ςt+sinh⁡ςtBδt±γ2s12dsinh⁡ςt-cosh⁡ςtNδt+dcosh⁡ςt+sinh⁡ςtBδt+Dt,with D(t) being a curve in L3 and ς′(t)=τδ(t).
From (76), we have D′t,D′t=η22t≥0. If η2(t)=0, then D(t) is constant. Thus, we assume η2(t)≠0. From (76), we have D′′t,D′′t=η2′(t)2≥0. If η2′(t)=0, η2(t) is constant and D(t) is a reparametrization of δ(t) at constant speed. If η2′(t)≠0, we reparametrize D(t) by the arc-length parameter u and call m(u)=Dtu to the reparametrized curve. In this case, one can check that d2m/du2 is null and(78)Nmu=1η2tuNδtu,Bmu=η2tuBδtu,τmu=-1η2tuη2tudu+τδtuη2tu.Then, (77) can be written as(79)ys,u≔xs,tu=mu+γ1s12dcosh⁡ς^u-sinh⁡ς^uNmu+dcosh⁡ς^u+sinh⁡ς^uBmu+γ2s12dsinh⁡ς^u-cosh⁡ς^uNmu+dcosh⁡ς^u+sinh⁡ς^uBmu,where mu is a spacelike curve in L3, {Nm,Bm} are the normal and binormal unit vectors of the Frenet frame along m(u), ς^(u)≔ς(t(u)), d/duς^(u)=τm(u) (torsion of m(u)), and γ(s) is the unique spacelike curve with curvature κ(s) (up to isometries) contained in the timelike plane Puo=u-1(to) described before. So this case falls under (A)(2).
Ifμ2(s)=-de-∫‍κ(s)ds in (73), then, taking now ς˙1(s)=-κ(s), one can repeat the argument following (76) and gets a similar conclusion. This concludes Case  1.
Case  2 (Lorentzian surface
r
=
1
). If r=1, we have that x:M12→L3 is a Lorentzian surface. For the sake of brevity, most of computations in this case will be omitted. Since the geodesics of the foliation are not null, we distinguish two cases according to the causal character of α.
Case  2.1 (
α
is spacelike). With respect to the local parametrization of M12 in a neighbourhood of p defined by(80)xs,t=βts,t∈I, the metric can be written as(81)g=-ds2+G2dt2.Now, we have r=1 and ε4=1 in the PDE system (49), (50), and (51), the second equation of which gives again(82)xt=Gs,tTδt=Gs,tBs,t.Defining δ(t) by δ′(t)=Tδ(t), one has that δ(t) is spacelike and, using again similar arguments to those applied in the previous case (Riemannian surface, r=0), one can verify after a long computation that the same two cases (A)(1) and (A)(2) of Proposition 4 are obtained also in this case.
Case  2.2 (
α
is timelike). Let us suppose now that α(t) is a timelike curve. Then, the expression of the metric in the local parametrization we are using is(83)g=ds2-G2dt2,and we have to use r=1 and ε4=-1 in the PDE system (49), (50), and (51), again the second equation of which gives(84)xt=Gs,tTδt=Gs,tBs,t,where δ(t) is such that δ′t=Tδ(t)=Bs,t is a unit timelike vector field. Therefore, since δ′t=Bs,t, δ′′ falls into the spacelike plane generated by T(s,t)=xs and ξ, because δ′,δ′′=0 and δ′′ is not null. Now,(85)xs=cos⁡θs,tNδ+sin⁡θs,tBδ,and one can use again similar arguments to those applied in case of a Riemannian surface for nonnull δ′′, substituting cosh and sinh by cos and sin, respectively, obtaining case (B) of Proposition 4.


## 4.2. Hashimoto Surfaces Foliated by Constrained Critical Geodesics of Type (3)

In 1906, da Rios [21] modeled the movement of a thin vortex filament in a viscous fluid by the motion of a curve propagating in R3 according to(86)∂x∂t=∂x∂s×∂2x∂s2,which is known as thelocalized induction equation, LIE (see [22]). For notation consistency, LIE is often to be written as(87)xt=xs×xss.In this section we are going to consider the evolution inL3 of a Frenet curve, γ(s), of rank 2 or 3 under LIE (87). Let x(s,t) describe the evolution of γ(s) under LIE and denote x(s,t)=γt(s), γ0(s)=γ(s), s∈(-ɛ,ɛ), and t∈(-δ,δ), where t represents the time evolution parameter. It is easy to show that if s is the proper time for γ(s), that is, the arc-length parameter, then so is the case for every t∈(-δ,δ). In fact, using (87) we have(88)∂∂tγ˙ts,γ˙ts=2det⁡∂x∂s,∂3x∂s3,∂x∂s=0;that is, 〈∂γt/∂s,∂γt/∂s〉 does not depend on “time” t, so since 〈∂γ0/∂s,∂γ0/∂s〉=〈∂γ/∂s,∂γ/∂s〉=〈T,T〉=ε1, then so is the case for every t∈(-δ,δ).From now on, we will assume thats is the arc-length parameter and that xss is nonnull everywhere. Then for any fix to we may consider the associated Frenet frame {T=γ˙,N,B}(s,to) on x(s,to)=γto(s) described in (16). We are going to assume also that x(s,t) defines an immersed surface in L3 which will be called aHashimoto surface (with initial condition γ) Sγ. For any given to, the curve x(s,to) will be referred to as avortex curve. If every vortex curve is a C4 closed curve, the Hashimoto surface will be calledHashimoto tube.Since our curvesγ are arc-length parametrized, LIE can be simplified in terms of thebinormal flow. To be more precise, using (16) and (17) in (87) we have(89)∂x∂t=∂x∂s×∂2x∂s2=T×DTds=T×ε2κN=ε2ε3κB.This means that if γ is a Frenet curve of rank 2 or 3 parametrized by arc-length and evolving under LIE (87) with x(s,0)=γ0(s)=γ(s), then it evolves by the binormal flow.Observe that for a Hashimoto surfaceSγ the filament evolution x(s,t) under LIE implies that the vortex curves (t-curves) x(s,to) are geodesics in Sγ and then x(s,t) gives a parametrization of Sγ of type (26) where, as a consequence of (89), the induced metric is expressed as in (27) with G=ε2ε3κ, with κ being the curvature of an orthogonal curve to the geodesic foliation of Sγ determined by x(s,t). Hence, one can see that the Gauss-Codazzi equations (37)–(39) and the PDE system (40)–(42) reduce, respectively, to(90)κt=-ε2ε32κsτ+κτs,(91)τt=ε2ε2κssκ-ε3τ2+12ε1κ2s,(92)xss=ε2κN,(93)xst=κsκxt-ε3κτN,(94)xtt=-ε1ε3κκsxs+ε2ε3-2κsτκ-τsxt+ε2κτ2-ε3κssN.Notice that if we were considering evolution under LIE in the standard Euclidean case, then εi=1, for i=1,2,3, and (90) and (91) would be the well-knownda Rios equations [21]. In other words, in the Riemannian case da Rios equations are nothing but the Gauss-Codazzi equation of Hashimoto surfaces expressed with respect to the geodesic coordinate system (26). By this reason, the Gauss-Codazzi equations of the Lorentzian case, (90) and (91), will be referred to as theLorentzian da Rios equations.Lorentzian Hashimoto surfaces have the following properties.Proposition 5.
With the previous notation, letSγ be a Hashimoto surface having by initial condition a Frenet curve of rank 2 or 3 in L3, γ(s), parametrized by proper time. Denote by x(s,t) the parametrization of Sγ determined by LIE (87). Then, one has the following. (1)
Ifs∈[0,1] and all vortex curves are C4-closed in [0,1], then ∫o1κ2ds and ∫o1τds are independent of t.(2)
If all vortex curves are planar, then they are elasticae in either a Riemannian or a Lorentzian plane. The corresponding Hashimoto surface is described in Proposition4 and if δ′′ is not null, then Sγ is either a right cylinder on a Lorentzian circle or a rotation surface shaped on a planar elastica γ of either R2 or L2.(3)
The initial vortex curveγ(s) evolves by rigid motions under LIE, if and only if it is an elastica in L3. As a consequence, a rank 3 elastica γ(s) in L3 evolves under LIE (by rigid motions) and the different positions of the vortex curve over time give foliation of the associated Hashimoto surface by Sγ-constrained elastic geodesics of type 3 in Proposition 3.Proof.
Assume thats∈[0,1] and that, for every t, x(s,t) is C4-closed in [0,1]. Then, using (90),(95)ddt∫01κ2s,tds=2∫01κtκds=-2ε2ε3∫01ddsκ2τds=0.Thus, ∫o1‍κ2ds attains the same value at every vortex curve. That ∫o1‍τds does not depend on time evolution follows trivially from (91).
Now, we want to prove2. If all vortex curves are planar, then τ=0 and, therefore, since they are not null, they must lie either in an Euclidean plane R2 or in a Lorentzian plane L2. Moreover, (90) implies that κ(s,t)=κ(s) does not depend on t and (91) gives ε2κss+1/2ε1κ3+μκ=0, μ∈R, which is precisely the equation for an elastica either in R2 (ε1=ε2=1) or in L2 (ε1ε2=-1) [6]. On the other hand, since τ=0, we have that B(s,t)=B(t) does not depend on s. If κ were constant, then vortex curves are circles (planar curves with constant curvature in L3) and, after differentiation of (89) with respect to t combined with the Frenet equations (16), we get that B(t) is a constant vector of L3. Hence, Sγ is a right circular cylinder. If κ is not constant, we combine (93) with (89) to obtain(96)κκδNδ=-ε1ε3κκsxs-ε2ε3κssκxss,where we are using the notation of Proposition 4 and assuming that δ′′ is not null. Differentiating (96) with respect to t and using (16) again, we obtain that κδ is constant and τδ=0. Hence, δ(t) is a planar circle. Assume first that δ is timelike. Then, Bδ is a spacelike constant vector that we may choose to be (0,0,1). Thus, δ can be parametrized as δ(s)=(1/kosinh⁡kot,1/kocosh⁡kot,0). Hence, by repeating the argument in the proof of Proposition 4, we have(97)xs,t=κskosinh⁡kot,κskocosh⁡kot,as,where κs2/ko2+as2=1 and the curve γ(s)=(κ(s),a(s)) is an elastica either in R2 or in L2. Thus Sγ is a rotation surface with profile curve γ. A similar conclusion can be achieved if δ is spacelike. This finishes item 2.
Finally, let us prove3. If γ evolves by rigid motion of L3, then κ(s,t)=κ(s) and τ(s,t)=τ(s) do not depend on t, so the left hand sides of (90) and (91) are zero. In particular, we get (98)0=2κsτ+κτs,0=2ε2κss-2ε3τ2κ+ε1κ3+μκ,μ∈R, which are the equations for elasticae in L3 [6]. In particular, γ is an elastica in L3.
Conversely, assume thatγ(s)=x(s,0) is an elastica in γ(s); then, the following vector field along γ(s),(99)Js=ε2ε3κs,0Bs,0,can be extended to a Killing field on L3, V [6]. Denote by {ϕt,t∈R} the 1-parameter group of isometries of L3 associated with V and consider y(s,t)≔ϕt(γ(s)). Then, since ϕt are isometries, we have γ(s)=y(s,0) and ε2ε3κs,tBs,t=yt(s,t), what implies that y(s,t)=x(s,t) and x(s,t) describes an evolution of γ(s)=x(s,0) under LIE by rigid motions.
As for the last part of statement3, if γ(s) is a rank 3 elastica in L3, then we already know that it evolves under LIE by rigid motions. Therefore, the different positions of the vortex curve over time, x(s,t), are also rank 3 elastica in L3. Trivially, elasticae in L3 are also Sγ-constrained elasticae; thus, x(s,t) gives a foliation of the associated Hashimoto surface, Sγ, by Sγ-constrained elastic geodesics of type 3 in Proposition 3.Remark 6.
Observe that these facts are an extension of the corresponding properties in the Riemannian setting. In fact, takingεi=1, i∈1,2,3, in the above proof, we see that these properties are also true for Hashimoto surfaces in R3. In this case, a somehow different proof of items (2) and (3) of Proposition 5 can be found in [23] and [24], respectively.
Also, in connection with item2 of Proposition 5, it should be mentioned that as it has been proved in [25], Hashimoto surfaces of revolution in R3 (resp., in L3) must be shaped on elasticae of R2 (resp., of L2) and, moreover, they provide congruence solutions of (87).



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