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Advances in Mathematical Physics
Volume 2015, Article ID 458178, 13 pages
http://dx.doi.org/10.1155/2015/458178
Research Article

Hypersurface Constrained Elasticae in Lorentzian Space Forms

1Department of Mathematics, Faculty of Science and Technology, Apartado 644, 48080 Bilbao, Spain
2Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu 702-701, Republic of Korea

Received 30 August 2015; Accepted 13 October 2015

Academic Editor: Pavel Kurasov

Copyright © 2015 Óscar J. Garay et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study geodesics in hypersurfaces of a Lorentzian space form , which are critical curves of the -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 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.

1. 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 , 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 , the problem of elastic curves in surfaces has a long history, but it is really well understood only when 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., [46]).

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, [7]. In particular, he obtained the Euler-Lagrange equation of , with being the curvature in of a curve , for variations of constrained to lie in with prescribed first-order boundary data. Different versions of this problem for surfaces in -space forms for a variety of curvature energies and boundary conditions have been considered in [811]. 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 , in a Lorentzian space form , 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 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 . Then, in Section 4, we restrict ourselves to the flat Minkowski space 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 with a rank 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 -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

Let be a Lorentzian manifold with metric and Levi-Civita connection . If is a smooth immersed curve in , will represent its velocity vector and the covariant derivative of a vector field along will be denoted by . A immersed curve in Lorentzian manifold is spacelike (resp., timelike; resp., light-like) if , (resp., , ; resp., , ). Of course, there exist curves whose causal character changes as 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 called proper time and usually denoted by .

For a nonnull immersed curve parametrized by arc-length , the first Frenet curvature, or simply the curvature, is defined as , where denotes the causal character of . A geodesic is a constant speed curve whose tangent vector is parallel propagated along itself, that is, a curve whose tangent, , satisfies the equation . Obviously, geodesics have zero curvatures. In this paper, geodesics will be called Frenet curve of rank . An immersed curve in a Lorentzian manifold is called a Frenet curve of rank , , if is the highest integer for which there exists an orthonormal frame defined along , , and nonnegative smooth functions on , , , (Frenet curvatures), such that the following equations are satisfied (Frenet-Serret equations):

where denotes the causal character of , . For a Frenet curve of rank , the Frenet curvatures of index higher than are considered to be zero, , , .

A complete, connected, simply connected, Lorentzian manifold with constant sectional curvature is called a Lorentzian space form . 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 completely determine the curve up to isometries. Moreover, given functions we can always construct a spacelike (resp., timelike) Frenet curve, parametrized by the arc-length, whose curvature functions are precisely the . Then, any local geometrical scalar defined along Frenet curves can always be expressed as a function of their curvatures and derivatives.

Now, let , , be a semi-Riemannian manifold isometrically immersed in a Lorentzian space form . For a given , the first normal space is defined to be the subspace of the normal space spanned by the vector valued second fundamental form of the immersion. A normal subbundle is called parallel if, for each section of and each tangent vector , the covariant derivative of in the direction of , with respect to the normal connection, remains in . The following result is basically known.

Proposition 1. Let , , be a semi-Riemannian manifold isometrically immersed in Lorentzian space form . Assume that has a parallel normal subbundle of rank containing the first normal space . Then, there exists a totally geodesic submanifold embedded in such that .

Remark 2. If , the result can be found in [12] for (see also [13], Theorem ). If , 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 , and of [15], Theorem . 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 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 . Spacelike (resp., timelike) geodesics can be characterized as those constant speed immersed curves which are minimizers (resp., maximizers) of the length functional , with 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 or depending on whether is spacelike or timelike, is the arc-length parameter, represents derivative with respect to , and is a real constant ( is supposed to act on a space of curves satisfying suitable initial and first-order boundary conditions, e.g., the space of curves defined in (4)). Clearly, geodesics of are elasticae. Moreover, elastic Frenet curves of Lorentzian space forms are known to lie fully in a totally geodesic submanifold of dimension at most , (for details, see [4]).

Now, let be a semi-Riemannian hypersurface of index isometrically immersed in a Lorentzian space form . In this case, can be either Riemannian, , or Lorentzian, . Assume that 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 . For simplicity, along this work, this problem will be referred to as the hypersurface constrained problem for or -constrained problem for . In contrast with what happens in the unconstrained problem, it turns out that a geodesic of is not necessarily a critical curve for the constrained problem. We first want to characterize geodesics of which are -constrained critical.

In order to derive first variation formulas for , we will use the following standard terminology (see [2, 5, 6] for details). For a nonnull immersed curve , with , we denote and by its unit tangent vector. Take a variation of , with . Associated with this variation, we have the variation vector field along the curve . We also write , , , , 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]where the Riemannian curvature tensor is defined by .

Choose two arbitrary points and vectors , , and consider the space of curveswhere denotes the derivative with respect to the parameter . We wish now to analyze the variational problem associated with energy (2) acting on .

Proposition 3. Let be a Frenet curve of rank which is geodesic of . Assume that is acting on . Then, is a critical point of (i.e., for the hypersurface constrained problem), if and only if one of the following cases occurs: (1)Rank of is ; that is, is a geodesic of .(2)Rank of is . That is, the torsion of vanishes, .(3) is a Frenet curve of rank satisfyingwhere is a constant and , are the two first Frenet curvatures of in . Moreover, in all the above cases lies fully in a totally geodesic submanifold of dimension , .

Proof. Let and take a variation of , by curves in 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 along in the direction of :where and stand for the Euler-Lagrange and Boundary operators, respectively, which are given byWe see that the initial and boundary conditions of the variation imply that the boundary term vanishes. Moreover, in a Lorentzian space form, the curvature tensor is given by so and (7) becomesSince and we are taking variations in , the variation field is tangent to along . So only the tangential part of affects the first variation formula (6) and is a critical point of , if and only ifwhere denotes tangential projection on .
Now, if rank of is , then it is a geodesic of and (11) trivially holds. Observe that if is a geodesic of , 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 , thenUsing (10), one sees that , for a certain function along the curve. But is normal to , since is a geodesic of the hypersurface, what means that and the curve is critical. On the other hand, the first normal space along is spanned also by and (13) shows that is a parallel normal subbundle of dimension . Hence, applying Proposition 1, we have that lies fully in a totally geodesic submanifold of dimension .
If rank of is , then the Frenet equations reduce toUsing (10), one sees that , for certain functions along the curve. Again, is normal to , because is a geodesic of the hypersurface, what means that . Hence, the curve is critical, if and only if . A computation gives , from where we obtain that must be constant along the curve. Moreover, if we consider the -dimensional normal bundle , we see that it contains the first normal space along , . In addition, the two last equations of (14) tell us that is parallel, so, applying again Proposition 1, we have that lies fully in a totally geodesic submanifold of dimension .
Finally, if rank of were , then, by using a similar argument, criticality of would imply also that should vanish. But a computation involving the Frenet equations and (10) would give , which contradicts that the rank is . 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 immersed in the Minkowski -space , which are critical for the surface constrained problem, must fall under one of the three cases described there. We want to study surfaces in foliated by such critical geodesics. The main goal of the next section is to determine all surfaces of locally foliated by critical geodesics of type of Proposition 3. Then, in the final part of the section we will give a method to construct surfaces of locally foliated by critical geodesics of type of Proposition 3.

4. Surfaces in Foliated by Surface Constrained Critical Geodesics

Consider the Minkowski -space , that is, the flat Lorentzian -space equipped with the metricwhere is the standard rectangular coordinate system. The standard metric (15) will be denoted by . As usual, the cross product of two vector fields , in , denoted by , is defined so that for any other vector field of .

Let be a unit speed nongeodesic curve contained in with nonnull velocity . If it also has nonnull acceleration , then is a Frenet curve of rank or and the classical standard Frenet frame along is given by , and is chosen so that . From now on, the first and second Frenet curvatures will be denoted by and will be referred to as the curvature and torsion of in , respectively. Then, the Frenet equations (1) can be written aswhere , , denotes the causal character of , , and , respectively, and the following relations hold:Notice that even if the rank of is , the binormal is still well defined and above formulas (16) still make sense when .

If had null acceleration , then we would consider the following frame along . Take and denote by the only light-like vector such that and . Again, the vectors are referred to as the unit normal and binormal vectors of , respectively. In this case, the “Frenet” equations arefor 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 -space 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 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 locally foliated by critical geodesics of type . 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 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 (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 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 , these and admit another foliation by geodesics of the third type of Proposition 3.

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

As it is also customary, for a surface 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, (), if the first fundamental form is positive definite; it is called timelike or Lorentzian, , if the first fundamental form is indefinite. In this section, for a given surface , , we denote by the Levi-Civita connection on associated with the metric (15) and by the Levi-Civita connection of the immersion . Let be the second fundamental form of in . From now on, , , , and represent vector fields tangent to and denotes a vector field normal to . Then, the formulas of Gauss and Weingarten are [13]Here, denotes the connection on the normal bundle of .

If we denote the Riemann curvature tensor of by , then the equations of Gauss and Codazzi are given, respectively, by [13]whereNow choosing an adapted local orthonormal frame in such that the vectors , are tangent to and is normal to in and denoting by the dual frame of , the Cartan connection forms are defined byfor , where is the causal character of . Then, and [13].

Let denote a surface in Lorentzian -space (if , we simply denote ) with local orientation determined by the normal vector . Take and let be an immersed nonnull curve ; that is, is a nonnull vector for all with causal character , such that is contained in a local chart around and . For any , take as a unit vector tangent to at so that form an orthonormal basis and consider the geodesic with initial data: and . For the local parametrization of in a neighbourhood of defined by, the coefficients of the metric are (reparametrizing the geodesics if needed) , , and , which, for simplicity, is denoted by . That is, with respect to this parametrization, the metric can be written as

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 havewhere subscripts and mean partial derivative with respect to and , respectively. Hence the Levi-Civita connection of is given byMoreover, combining these equations and (21) we obtain that the Gaussian curvature of is given by

If , , were also a geodesic in , then would be a ruled surface. So assume that is not a geodesic in , then is not null (upper meaning derivative with respect to ) and the unit Frenet normal to is parallel to the unit normal to , 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 :where is the unit normal to . Then, combining (19), (20), (24), (25), and (16), one getswhere and denote the curvature and torsion of the curves .

The second fundamental form can be considered as a quadratic form given by ; therefore, we obtain from (35) thatwith 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

Using Gauss formula (19) and the expression for Levi-Civita connection (29), one sees that the immersion satisfies the following PDE system:

Conversely, observe that (37)–(39) are the compatibility conditions for the PDE system (40)–(42). Thus, given functions , , , and smoothly defined on a connected domain and satisfying (37)–(39), there exists a solution of (40)–(42) determining a smooth immersion of a surface in whose metric and second fundamental form are given by (27) and (36), respectively. This surface is foliated by geodesics having and as curvature and torsion. If, in addition, is simply connected, then the immersion would be unique (up to rigid motions in ). The regularity assumptions made before on the functions , , , and can be significantly relaxed in several ways. For instance, if they were of class , then we would have an immersion which would be . For weaker regularity conditions, one may see [19, 20], but here we restrict ourselves to the smooth case.

A nonnull unit speed curve of with lies in an affine plane. From now on, a curve with is going to be called a planar curve. We want to prove the following result.

Proposition 4. Let be a nonnull arc-length parametrized curve in , 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 the normal plane to at . Suppose first that is spacelike and take any nonnull arc-length parametrized curve in the timelike plane , . Then, one has the following.If is nonnull, define the surface given bywhere satisfies , denotes the torsion of , and is the causal character of . Then, the immersion given in (43) defines a surface of foliated by planar geodesics.If is null, consider the surface defined bywhere , , and is the torsion of . Then, the immersion given in (44) defines a surface of foliated by planar geodesics of .Moreover, in both cases, the pseudo-Riemannian character of the surface is determined by that of ; that is, is Riemannian (resp., Lorentzian) if and only if is spacelike (resp., timelike).Assume now that is timelike. Take any nonnull arc-length parametrized curve in the spacelike plane . Then,where and is the torsion of . Then, the immersion given in (45) defines a Lorentzian surface of foliated by planar geodesics of . Conversely, locally, any surface in 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 defined in (43), (44), and (45) gives a foliation of 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 is a surface in foliated by planar geodesics which is not a ruled surface. Consider a curve cutting orthogonally to the planar geodesic foliation and parametrize locally as in (26). Take and let be an immersed Frenet curve such that is contained in a local chart around and and is perpendicular to the planar geodesic foliation. Since , then (37)–(39) reduce toand (40), (41), and (42) toSince our curves , , are not geodesics in the Lorentz space , from (46) we have that depends only on . From now on, differentiation of one-variable functions will be denoted by and , respectively.
Case  1 (Riemannian surfaces: , ). Since the surface is Riemannian satisfying (27), we must have , , , and . From (50) we have , for some , , which combined with giveswhere , for some unit speed spacelike curve in ; that is, , the binormal of , is constant along the curve, as we already know since .
If were a geodesic in , then would be a unit constant vector and would imply that lies in a plane perpendicular to , . Hence, and combining (49), (50), (51), and (52), we obtain 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.
Case  1.1 ( and is a Frenet curve: is not a null vector). From and (52), we havewhere represents the Frenet frame along . Denote by , the causal characters of and , respectively. Since is spacelike, we must have .
Assume first and ; that is, is spacelike. Then, impliesfor some function . Now, combining this equation with (50), (52), and (53) and using the Frenet equations for , (16), one haswhere , stand for the curvature and torsion of . Hence, , with . Moreover, from (17), (49), and (54), we have Then, from the second equation of (55), we getOn the other hand, a combination of (54) and (57) results inDefineso that is the only (up to isometries) spacelike curve contained in the timelike plane with curvature . Then, (58) giveswhere is a curve in . Finally, using (60) along with (52), (55), and (57) we obtain after some computationswhere , . Observe that since is not a geodesic. Hence, from (61) we have , since , are spacelike vectors. Thus, if we reparametrize by the arc-length parameter and calling the reparametrized curve , one can check that is nonnull andwhere the subscript is meant to denote the geometric elements associated with . Hence, (60) can be written aswhere is a spacelike curve in , are the normal and binormal unit vectors of the Frenet frame along , is the torsion of , and is the unique spacelike curve with curvature (up to isometries) contained in the timelike plane described before. So this case falls under (A)(1).
If now we assume that and (i.e., is timelike), then analogous arguments lead toSo we arrive to similar conclusions by defining , . Thus, this case is also covered in (A)(1).
Case  1.2 ( and is a null vector). From and (52), we havewhere represents the frame along defined in (18). Then, impliesNow, combining this equation with (52) and (65) and using the Frenet equations for , (18), one hasWith integration, we obtainThen, combining (47) and (48) one hasor equivalentlyHence, from (70) and (68) one hasTherefore,for a certain constant . Observe that ; otherwise and would depend only on and would be zero. This would contradict (49) since we are assuming . Thus, by differentiating (65) and using (68), (71), and , one gets that in this case. Then, solving (72) we haveso (68) giveswith . Combining (65), (67), and (68), we obtainwhere is a curve in . Then, (52) givesAssume first that in (73). Now, defining by , consider the planar curve , in the timelike plane with coordinates and with respect to the frames and . That is, is the only (up to isometries) spacelike curve contained in the timelike plane with curvature . Then, (75) can be written in the following way:with being a curve in and .
From (76), we have . If , then is constant. Thus, we assume . From (76), we have . If , is constant and is a reparametrization of at constant speed. If , we reparametrize by the arc-length parameter and call to the reparametrized curve. In this case, one can check that is null andThen, (77) can be written aswhere is a spacelike curve in , are the normal and binormal unit vectors of the Frenet frame along , , (torsion of ), and is the unique spacelike curve with curvature (up to isometries) contained in the timelike plane described before. So this case falls under (A)(2).
If in (73), then, taking now , one can repeat the argument following (76) and gets a similar conclusion. This concludes Case  1.
Case  2 (Lorentzian surface ). If , we have that 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 .
Case  2.1 ( is spacelike). With respect to the local parametrization of in a neighbourhood of defined by, the metric can be written asNow, we have and in the PDE system (49), (50), and (51), the second equation of which gives againDefining by , one has that