Hypersurface Constrained Elasticae in Lorentzian Space Forms

We study geodesics in hypersurfaces of a Lorentzian space formM 1 (c), which are critical curves of theM 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 geodesicM 1 (c) and that theymust be of three different types. Finally, we consider the classification of surfaces in the Minkowski 3-space foliated by critical geodesics.


Introduction
Following a classical D. Bernoulli's model, a curve immersed in a Riemannian manifold  :  →   is called an elastic curve (or simply an elastica) if it is a minimum, or, more generally, a critical point, of the bending energy ∫   2   , where   denotes the geodesic curvature of  in   .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  = 2, the problem of elastic curves in surfaces has a long history, but it is really well understood only when  2 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][5][6]).
On the other hand, if  :  →   ⊂   () is a curve on a submanifold   immersed in a real -space form of constant curvature ,   (), one may wish to analyze the critical curves for the bending energy of the curve in   (), but for variations of  constrained to lie on the submanifold.In this paper, this problem will be referred to as the elastica constrained problem.The constrained problem was first considered by Santaló in the context of Euclidean surfaces,  = 0 [7].In particular, he obtained the Euler-Lagrange equation of ∫  2 , with  being the curvature in R 3 of a curve  :  →  2 ⊂ R 3 , for variations of  constrained to lie in  2 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][9][10][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 hypersurfaces    ,  = 0, 1, in a Lorentzian space form  +1 1 (), 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    , which are critical for the elastic energy of where   fl ⟨  (),   ()⟩ denotes the causal character of   (),  ∈ {1, . . ., }.For a Frenet curve of rank  < , the Frenet curvatures of index higher than  − 1 are considered to be zero,   () = 0,  ∈ ,  ≤  ≤  − 1.
A complete, connected, simply connected, Lorentzian manifold with constant sectional curvature  is called a Lorentzian space form  + 1 ().The fundamental theorem for Frenet curves of rank  tells us that, in a Lorentzian space form, the causal characters of the Frenet frame and the Frenet curvatures  1 , . . .,  −1 completely determine the curve up to isometries.Moreover, given functions  1 , . . .,  −1 we can always construct a spacelike (resp., timelike) Frenet curve, parametrized by the arc-length, whose curvature functions are precisely the  1 , . . .,  −1 .Then, any local geometrical scalar defined along Frenet curves can always be expressed as a function of their curvatures and derivatives.Now, let  :    →  + 1 (),  ∈ {0,1}, be a semi-Riemannian manifold isometrically immersed in a Lorentzian space form  + 1 ().For a given  ∈    , the first normal space N 1 () 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 called parallel if, for each section  of N and each tangent vector  ∈    , the covariant derivative of  in the direction of , with respect to the normal connection, remains in N. The following result is basically known.Remark 2. If  = 0, the result can be found in [12] for N 1 = N 2 (see also [13], Theorem 2.6).If  = −1, a proof is given in [14].A general proof for any  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].

Hypersurface Constrained Elasticae
In this section, we assume that all our curves are nonnull with nonnull acceleration.In other words, we assume that   () and   / are not light-like vectors along the curve.We use () for the arc-length reparametrization of () and in such a case the velocity vector is denoted by γ () = ()/.Spacelike (resp., timelike) geodesics can be characterized as those constant speed immersed curves which are minimizers (resp., maximizers) of the length functional Υ() fl ∫  √ 1 ⟨  ,   ⟩, with  1 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 the bending energy functional: where  2 = 1 or −1 depending on whether  γ / is spacelike or timelike,  is the arc-length parameter, γ represents derivative with respect to , and  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   1 are elasticae.Moreover, elastic Frenet curves of Lorentzian space forms   1 () are known to lie fully in a totally geodesic submanifold of dimension at most 3,  3  1 () (for details, see [4]).Now, let  :  −1  →   1 () be a semi-Riemannian hypersurface of index  isometrically immersed in a Lorentzian space form   1 ().In this case,  −1  can be either Riemannian,  = 0, or Lorentzian,  = 1.Assume that  :

𝑟
is a smooth immersed curve contained in the hypersurface.We are interested in those curves  of the hypersurface which are critical points of the bending energy (2) for variations contained in  −1  .For simplicity, along this work, this problem will be referred to as the hypersurface constrained problem for F or  −1  -constrained problem for F. In contrast with what happens in the unconstrained problem, it turns out that a geodesic of  −1  is not necessarily a critical curve for the constrained problem.We first want to characterize geodesics of  −1  which are  −1  -constrained critical.In order to derive first variation formulas for F, we will use the following standard terminology (see [2,5,6] for details).For a nonnull immersed curve  : I →  −1  , with I fl [0, 1], we denote V() = √ 1 ⟨  ,   ⟩() and by  = () its unit tangent vector.Take a variation of , Γ = Γ(, ) : [0, 1] × (−, ) →  −1  with Γ(, 0) = ().Associated with this variation, we have the variation vector field  = () fl (Γ/)(, 0) along the curve ().We also write  = (, ) = (Γ/)(, ),  = (, ), V = V(, ),  = (, ), and so forth, with the obvious meanings.Let  denote the arc-length, and put (, ), (, ), and so forth, for the corresponding reparametrizations.The following formulas can be computed as in [2,5] [, ] = − 1 ⟨∇  , ⟩ , where the Riemannian curvature tensor  is defined by Choose two arbitrary points   ∈  −1  and vectors V  ∈     −1  ,  ∈ {0, 1}, and consider the space of curves where (/)() denotes the derivative with respect to the parameter  ∈ I.We wish now to analyze the variational problem associated with energy (2) acting on Ω.
(3)  is a Frenet curve of rank 3 satisfying where  ∈ R is a constant and  1 ,  2 are the two first Frenet curvatures of  in   1 ().Moreover, in all the above cases  lies fully in a totally geodesic submanifold   ⊂   1 () of dimension  =  , 1 ≤  ≤ 3.
Proof.Let  ∈ Ω and take a variation Γ = Γ(, ) : [0, 1] × (−, ) →  −1  of , Γ(, 0) = () 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  in the direction of : where E() and B(, ) stand for the Euler-Lagrange and Boundary operators, respectively, which are given by We see that the initial and boundary conditions of the variation imply that the boundary term [B(, )] 1 0 vanishes.Moreover, in a Lorentzian space form, the curvature tensor is given by  (, )  =  (⟨, ⟩  − ⟨, ⟩ ) , so (/, ) =  1 (/) and ( 7) becomes Since  ⊂  −1  and we are taking variations in  −1  , the variation field  is tangent to  −1  along .So only the tangential part of E affects the first variation formula (6) and  is a critical point of F, if and only if tan (E ()) = 0, (11) where tan( ) denotes tangential projection on  −1  .Now, if rank of  is 1, then it is a geodesic of   1 () and ( 11) trivially holds.Observe that if  is a geodesic of   1 (), then it is a critical curve of the bending energy for any variation and so is the case for constrained variations.
If rank of  is 2, then Using (10), one sees that E() =  2 , for a certain function  along the curve.But  2 is normal to  −1  , since  is a geodesic of the hypersurface, what means that tan(E()) = 0 and the curve is critical.On the other hand, the first normal space N 1 along  is spanned also by  2 and (13) shows that N 1 is a parallel normal subbundle of dimension 1.Hence, applying Proposition 1, we have that  lies fully in a totally geodesic submanifold  2 ⊂   1 () of dimension 2. If rank of  is 3, then the Frenet equations reduce to Using (10), one sees that , from where we obtain that  2  1  2 must be constant along the curve.Moreover, if we consider the 2-dimensional normal bundle N = span{ 2 ,  3 }, we see that it contains the first normal space along , N 1 = span{ 2 }.In addition, the two last equations of (14) tell us that N is parallel, so, applying again Proposition 1, we have that  lies fully in a totally geodesic submanifold  3 ⊂   1 () of dimension 3. Finally, if rank of  were 4, then, by using a similar argument, criticality of  would imply also that E() =  2  2 +  3  3 +  4  4 should vanish.But a computation involving the Frenet equations and (10) would give  4 =  1  2  3 = 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 Proposition 3 that geodesics of a surface  2  immersed in the Minkowski 3-space L 3 , which are critical for the surface constrained problem, must fall under one of the three cases described there.We want to study surfaces in L 3 foliated by such critical geodesics.The main goal of the next section is to determine all surfaces of L 3 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 L 3 locally foliated by critical geodesics of type (3) of Proposition 3.

Surfaces in L 3 Foliated by Surface Constrained Critical Geodesics
Consider the Minkowski 3-space L 3 , that is, the flat Lorentzian 3-space R 3 equipped with the metric where ( 1 ,  2 ,  3 ) is the standard rectangular coordinate system.The standard metric (15) will be denoted by ⟨⋅, ⋅⟩.
As usual, the cross product of two vector fields ,  in L 3 , denoted by  × , is defined so that ⟨ × , ⟩ = det(, , ) for any other vector field  of L 3 .
Let  be a unit speed nongeodesic curve contained in L 3 with nonnull velocity γ = .If it also has nonnull acceleration  γ /, then  is a Frenet curve of rank 2 or 3 and the classical standard Frenet frame along  is given by { = γ ,  = ( 2 / 1 ) ∇ , }, and  is chosen so that det(, , ) = 1.From now on, the first and second Frenet curvatures { 1 ,  2 } will be denoted by {, } and will be referred to as the curvature and torsion of  in L 3 , respectively.Then, the Frenet equations 2 can be written as where   , 1 ≤  ≤ 3, denotes the causal character of , , and , respectively, and the following relations hold: Notice that even if the rank of  is 2, the binormal  =  3 × is still well defined and above formulas ( 16) still make sense when  = 0.
If  had null acceleration  γ /, then we would consider the following frame along .Take () = ∇  and denote by () the only light-like vector such that ⟨, ⟩ = 1 and ⟨, ⟩ = 0. Again, the vectors {(), ()} are referred to as the unit normal and binormal vectors of , respectively.In this case, the "Frenet" equations are for a certain function () which will be also called torsion.
There is no definition for curvature in this case.
On the other hand, a ruled surface  in 3-space L 3 is defined by the property that it admits a parametrization (, ) = () + (), where () is a connected piece of a regular curve and () is a nowhere vanishing vector field along the curve.Thus, rulings ( = ) of  are geodesics of L 3 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 L 3 locally foliated by critical geodesics of type (2).By a Lorentzian rotation around an axis is meant Lorentzian transformation leaving a straight line (the axis) pointwise fixed.Rotation surfaces are those surfaces in L 3 which are invariant by the 1-parameter group of the Lorentzian isometries which leave a straight line  (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  = 0 (second type of Proposition 3).Now, for a given ruled surface (, ) = () + (), the curve () is called a base curve and () a director curve.In particular, the ruled surface is said to be cylindrical if the director curve () =  is constant and noncylindrical otherwise.If, in addition,  is perpendicular to   (), then (, ) = () +  is called a right cylinder on ().Now, assume that the base curve () is a null (light-like) curve in L 3 with Cartan frame {, , }; then, (, ) = () + () is a Lorentz surface which is called a null -scroll over (), [18].On the other hand, a Frenet curve is called a Frenet 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,  , , while degenerate Frenet helices are geodesics in flat -scrolls,  , [5].Hence, in addition to the foliation by geodesics of L 3 , these  , and  , admit another foliation by geodesics of the third type of Proposition 3.

Surfaces in L 3 Foliated by Surface Constrained Critical
Geodesics of Type (2).As it is also customary, for a surface  2   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 called spacelike or Riemannian,  2 ( = 0), if the first fundamental form is positive definite; it is called timelike or Lorentzian,  2  1 , if the first fundamental form is indefinite.In this section, for a given surface  :  2  → L 3 ,  ∈ {0, 1}, we denote by ∇ the Levi-Civita connection on L 3 associated with the metric ⟨, ⟩ (15) and by ∇ the Levi-Civita connection of the immersion ( 2  , ).Let ℎ be the second fundamental form of  2  in L 3 .From now on, , , , and  represent vector fields tangent to  2   and  denotes a vector field normal to  2  .Then, the formulas of Gauss and Weingarten are [13] ∇  = ∇   + ℎ (, ) , Here,  ⊥ denotes the connection on the normal bundle of  2  .
Let  :  2  → L 3 denote a surface in Lorentzian 3space L 3 (if  = 0, we simply denote  2  =  2 ) with local orientation determined by the normal vector .Take  ∈  2   and let  :  →  2  be an immersed nonnull curve (); that is, / is a nonnull vector for all  ∈  with causal character  4 , such that () is contained in a local chart around  and (0) = .For any   ∈ , take V  as a unit vector tangent to  2  at (  ) so that {(/)(  ), V  , (  )} form an orthonormal basis and consider the geodesic    () with initial data:    (0) = (  ) and (   /)(0) = V  .For the local parametrization of Using the metric coefficients   , 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 where subscripts  and  mean partial derivative with respect to  and , respectively.Hence the Levi-Civita connection of  2  is given by If   (), ∀, were also a geodesic in L 3 , then  2  would be a ruled surface.So assume that   () is not a geodesic in L 3 , then ∇ β  β  = β  () is not null (upper β meaning derivative with respect to ) and the unit Frenet normal to   () is parallel to the unit normal to  2  , for  lying in a certain interval   .Let us denote by {(, ), (, ), (, )} the Frenet frame of   () as described in (16) and choose the following local adapted frame on  2  : where  is the unit normal to  2  .Then, combining ( 19), ( 20), ( 24), (25), and ( 16), one gets where (, ) and (, ) denote the curvature and torsion of the curves   ().
A nonnull unit speed curve of L 3 with  = 0 lies in an affine plane.From now on, a curve with  = 0 is going to be called a planar curve.We want to prove the following result.Proposition 4. Let  :  1 → L 3 be a nonnull arc-length parametrized curve () in L 3 , and let {  (),   (),   ()} denote either its Frenet frame given in (16) if   () is nonnull or the frame given in (18) if   () is null.We also denote by    fl span{  (  ),   (  )} the normal plane to () at   ∈  1 .
Conversely, locally, any surface  2  in L 3 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 curves  a (, ⋅) defined in ( 43), (44), and (45) gives a foliation of (,  a ) 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 .Their principal curvature is (), where  denotes the causal character of () and () is the curvature of .
For the converse, assume that  2  is a surface in L 3 foliated by planar geodesics which is not a ruled surface.Consider a curve  cutting orthogonally to the planar geodesic foliation and parametrize  2  locally as in (26).Take  ∈  2   and let  :  →  2  be an immersed Frenet curve () such that () is contained in a local chart around  and (0) =  and is perpendicular to the planar geodesic foliation.Since  = 0, then (37)-(39) reduce to and ( 40), (41), and (42) to Since our curves   (),  ∈   , are not geodesics in the Lorentz space  ̸ = 0, from (46) we have that () depends only on .From now on, differentiation of one-variable functions (), () will be denoted by φ () = ()/ and   () = ()/, respectively.
If () were a geodesic in L 3 , then   =  would be a unit constant vector and ⟨  ,   ⟩ = 0 would imply that   Advances in Mathematical Physics lies in a plane perpendicular to ,   .Hence,   ∈   and combining (49), ( 50), (51), and (52), we obtain   = 0 which means that (, ) = ()+(), with () being a unit speed curve in   and   () = ().Thus, our surface would be a right circular cylinder,   , shaped on a planar spacelike curve  contained in   .The causal character of   is determined by that of .So we may assume () is not a geodesic.Now, we distinguish two cases.
∈ , the metric can be written as Now, we have  = 1 and  4 = 1 in the PDE system (49), (50), and (51), the second equation of which gives again Defining () by   () =   (), one has that () is spacelike and, using again similar arguments to those applied in the previous case (Riemannian surface,  = 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 ( is timelike).Let us suppose now that () is a timelike curve.Then, the expression of the metric in the local parametrization we are using is and we have to use  = 1 and  4 = −1 in the PDE system (49), (50), and (51), again the second equation of which gives where () is such that   () =   () = (, ) is a unit timelike vector field.Therefore, since   () = (, ),   falls into the spacelike plane generated by (, ) =   and , because ⟨  ,   ⟩ = 0 and   is not null.Now, and one can use again similar arguments to those applied in case of a Riemannian surface for nonnull   , substituting cosh and sinh by cos and sin, respectively, obtaining case (B) of Proposition 4.

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 R 3 according to which is known as the localized induction equation, LIE (see [22]).For notation consistency, LIE is often to be written as In this section we are going to consider the evolution in L 3 of a Frenet curve, (), of rank 2 or 3 under LIE (87).Let (, ) describe the evolution of () under LIE and denote (, ) =   (),  0 () = (),  ∈ (−, ), and  ∈ (−, ), where  represents the time evolution parameter.It is easy to show that if  is the proper time for (), that is, the arc-length parameter, then so is the case for every  ∈ (−, ).In fact, using (87) we have that is, ⟨  /,   /⟩ does not depend on "time" , so since ⟨ 0 /,  0 /⟩ = ⟨/, /⟩ = ⟨, ⟩ =  1 , then so is the case for every  ∈ (−, ).
From now on, we will assume that  is the arc-length parameter and that   is nonnull everywhere.Then for any fix   we may consider the associated Frenet frame { = γ , , }(,   ) on (,   ) =    () described in (16).We are going to assume also that (, ) defines an immersed surface in L 3 which will be called a Hashimoto surface (with initial condition )   .For any given   , the curve (,   ) will be referred to as a vortex curve.If every vortex curve is a  4 closed curve, the Hashimoto surface will be called Hashimoto tube.
Since our curves  are arc-length parametrized, LIE can be simplified in terms of the binormal flow.To be more precise, using ( 16) and ( 17 Observe that for a Hashimoto surface   the filament evolution (, ) under LIE implies that the vortex curves (curves) (,   ) are geodesics in   and then (, ) gives a parametrization of   of type (26) where, as a consequence of (89), the induced metric is expressed as in (27) with  =  2  3 , with  being the curvature of an orthogonal curve to the geodesic foliation of   determined by (, ).Hence, one can see that the Gauss-Codazzi equations (37)-(39) and the PDE system (40)-(42) reduce, respectively, to Notice that if we were considering evolution under LIE in the standard Euclidean case, then   = 1, for  = 1, 2, 3, and (90) and ( 91) would be the well-known da 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 the Lorentzian da Rios equations.
Lorentzian Hashimoto surfaces have the following properties.
As for the last part of statement 3, if () is a rank 3 elastica in L 3 , then we already know that it evolves under LIE by rigid motions.Therefore, the different positions of the vortex curve over time, (, ), are also rank 3 elastica in L 3 .Trivially, elasticae in L 3 are also   -constrained elasticae; thus, (, ) gives a foliation of the associated Hashimoto surface,   , by  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, taking   = 1,  ∈ 1, 2, 3, in the above proof, we see that these properties are also true for Hashimoto surfaces in R 3 .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 item (2) of Proposition 5, it should be mentioned that as it has been proved in [25], Hashimoto surfaces of revolution in R 3 (resp., in L 3 ) must be shaped on elasticae of R 2 (resp., of L 2 ) and, moreover, they provide congruence solutions of (87).