Geometry of the Solutions of Localized Induction Equation in the Pseudo-Galilean Space

The localized induction equation (LIE), also called the vortex filament equation or the Betchov-Da Rios equation, is an idealized model of the evolution of the centerline of a thin vortex tube in a three-dimensional inviscid incompressible fluid. The connection of LIE with the theory of solitons was discovered by Hasimoto proving that the solutions of LIE are related to solutions of the cubic nonlinear Schrödinger equation, which is well known to be an equation with soliton solutions. For details, see [1–5]. The soliton surface associated with the nonlinear Schrödinger equation is called a Hasimoto surface. The geometric properties of such surfaces are investigated in [4, 6, 7]. Bymotivating the fact that the study of theHasimoto surfaces can be interesting in the pseudo-Galilean space which is one of the Cayley-Klein spaces, we are mainly interested in the geometric properties of these surfaces in the pseudo-Galilean space. Let (M, g) be a 3-dimensional Riemannian manifold and ∇ the Levi-Civita connection with respect to g. Note that the cross product of two vector fields X,Y on M may be defined as


Introduction
The localized induction equation (LIE), also called the vortex filament equation or the Betchov-Da Rios equation, is an idealized model of the evolution of the centerline of a thin vortex tube in a three-dimensional inviscid incompressible fluid.The connection of LIE with the theory of solitons was discovered by Hasimoto proving that the solutions of LIE are related to solutions of the cubic nonlinear Schrödinger equation, which is well known to be an equation with soliton solutions.For details, see [1][2][3][4][5].
The soliton surface associated with the nonlinear Schrödinger equation is called a Hasimoto surface.The geometric properties of such surfaces are investigated in [4,6,7].By motivating the fact that the study of the Hasimoto surfaces can be interesting in the pseudo-Galilean space which is one of the Cayley-Klein spaces, we are mainly interested in the geometric properties of these surfaces in the pseudo-Galilean space.
Let (, ) be a 3-dimensional Riemannian manifold and ∇ the Levi-Civita connection with respect to .Note that the cross product of two vector fields ,  on  may be defined as  ( × , ) = V  (, , ) , where V  is the volume form of the manifold and , ,  ∈ Γ().Denote by r = r(, ) the vortex filament; then LIE or the Betchov-Da Rios filament equation is On the other hand, the Galilean space which can be defined in three-dimensional projective space  3 (R) is the space of Galilean Relativity.The geometries of Galilean space and pseudo-Galilean space have similarities but, of course, are different.For study of surfaces in the pseudo-Galilean space, we refer to Šipuš and Divjak's series of works [8][9][10][11][12][13].
In classical geometry of surfaces, it makes sense to classify the surfaces having null curvature.In particular, a surface is said to be developable if it has null Gaussian curvature.In this case the surface can be flattened onto a plane without distortion.We remark that cylinders and cones are examples of developable surfaces, but the spheres are not under any metric.
There exist significant applications of the results obtained on the surfaces of null curvature in different fields, for example, in microeconomics.When the graphs of production functions in microeconomics have null Gaussian curvature, one can realize a "good" analysis of isoquants by projections, without losing essential information about their geometry

Preliminaries
The pseudo-Galilean space G 1  3 is one of the Cayley-Klein spaces with absolute figure that consists of the ordered triple {, , }, where  is the absolute plane in the threedimensional real projective space  3 (R),  is the absolute line in , and  is the fixed hyperbolic involution of points of  [8,9,13].
Let  be an admissible curve given by () = ( 1 (),  2 (),  3 ()) in G 1  3 ; that is, it has no isotropic tangent vectors for ∀.Note that an admissible curve has nonzero curvature.Then its curvature and torsion functions are, respectively, defined by where γ  () =   / and so forth, 1 ≤  ≤ 3.If  is an admissible curve parameterized by the arc length , it is of the form The associated Frenet frame field of  is the trihedron {T, N, B} such that where In this sense, the Frenet formulas for the curve  are [17,18] Let  2 be a surface in the pseudo-Galilean space G 1 3 parameterized by or, in matrix form, where Advances in Mathematical Physics 3 Let us define the function  as Thus a side tangential vector is defined by which can be a spacelike isotropic or a timelike isotropic vector.The unit normal vector field U of  2 is given by The second fundamental form II of  2 and its coefficients are defined by where  = U⋅U = −S⋅S = ±1.Two types of admissible surfaces can be distinguished: spacelike surfaces having timelike unit normals ( = −1) and timelike ones having spacelike unit normals ( = 1).The third fundamental form of  2 is where The Gaussian curvature and the mean curvature of  2 are, respectively, defined by A surface in G 1 3 is said to be minimal if its mean curvature vanishes.

Hasimoto Surfaces with Null
Curvature in G where r :  ⊂ R 2 → G 1 3 is smooth and regular mapping such that  2 = r().
In order to parameterize by arc length the curve r(, ) for ∀ ∈   , the parameterization of the surface in G 1  3 may be chosen as follows: where  = () is a smooth function of one variable on   .After using the pseudo-Galilean cross product given by ( 5) and ( 21), it can be easily seen that the function () is a constant function.In this sense, we get the parameterization of a Hasimoto surface  2 in G 1 3 as r (, ) = ( + ,  2 (, ) ,  3 (, )) (23) for arbitrary constant .Note that the Hasimoto surface given by ( 23) is always admissible; that is, it has no pseudo-Euclidean tangent planes, if ( 2 )  ̸ = 0. Now, let  2 be a Hasimoto surface given by ( 23) in G 1  3 .Denote the associated Frenet frame field of the curve  = (, ) for ∀ ∈   by {T, N, B}.Then, the derivative of T with respect to  has the expression for some smooth functions  1 ,  2 on .By ( 21) and (24), we get Moreover, applying the compatibility condition T  = T  yields that In similar ways, we deduce

Advances in Mathematical Physics
The compatibility conditions N  = N  and B  = B  give Summing up, we have the following result.
Lemma 1.Let  2 be a Hasimoto surface given by ( 23) in G 1 3 .Then the following equations hold: We plot the surface  2 as in Figure 1.
We plot the surface  2 as in Figure 2. Theorem 4. Let  2 be a Hasimoto surface given by ( 23) in G 1 3 .Then (i)  2 has null Gaussian curvature in G 1  3 if and only if (  /) + ( − 1) 2 = 0; (ii)  2 is a minimal Hasimoto surface in G 1  3 if and only if   +  2 = 0, where  and  are, respectively, the curvature and the torsion of the curve  = (, ) for ∀ ∈   .
Proof.Let  2 be a Hasimoto surface given by ( 23) in G 1  3 .Then the unit normal vector field U of  2 is where N(, ) is the principal normal vector field of the curve r = r(, ) for ∀ ∈   .Thus, we have From ( 19) and ( 35), we deduce that which gives the first statement.
Next by substituting (35) into (20), we obtain that  2 is a minimal Hasimoto surface if and only if Thus the proof is completed.
Gheorghe Tzitzeica (1873-1939) introduced a class of curves, nowadays called Tzitzeica curves, and a class of surfaces of the 3-dimensional Euclidean space, called Tzitzeica surface.
A Tzitzeica surface is a spatial surface for which the ratio of its Gaussian curvature  and the distance  from the origin to the tangent plane at any arbitrary point of the surface satisfy  =  4 for a constant .This class of surface is of great interest, having important applications both in mathematics and in physics (see [15]).The relation between Tzitzeica curves and surfaces is the following: For a Tzitzeica surface with negative Gaussian curvature, the asymptotic lines are Tzitzeica curves [19][20][21].
It is easy to prove that the tangent plane at an arbitrary point of any Hasimoto surface passes through the origin of G 1  3 .That is why all Hasimoto surfaces having null Gaussian curvature satisfy Tzitzeica condition in G 1  3 .Therefore the following result can be given without proof.
On the other hand, the third fundamental form III of a surface in G 1  3 may be introduced in the analogous way as in Euclidean space.Let  2 be a surface and U its unit normal vector in G 1 3 .If  2 is spacelike (timelike) in G 1 3 , then the end points of associated position vectors of U lie on a unit spacelike sphere  2 −  2 = 1 (unit timelike sphere  2 −  2 = 1).The mapping obtained in such a way is called the Gauss mapping or the spherical mapping in G 1  3 .The set of all end points of U is called the spherical image of  2 in G 1 3 .In this sense, the third fundamental form is indeed the first fundamental form of the spherical image (cf.[13]).Thus the following result for the Hasimoto surfaces in G 1  3 can be given.
Proof.Let us consider the Hasimoto surface  2 given by ( 23) in G 1  3 .Then from (35) we have  2 = 0 as one of the coefficients of the first fundamental form of  2 .This immediately implies from (18) that  11  22 −  2  12 = 0, which completes the proof.

Curves on Hasimoto Surfaces in G 1 3
There exists a frame field, also called the Darboux frame field, for the curves lying on surfaces apart from the Frenet frame field.For details, see [22,23].Let  be a curve lying on the surface  2 with unit normal vector field U.By taking T =  * (/) one can get a new frame field {T, T × U, U} which is the Darboux frame field of  with respect to  2 .On the other hand, the second derivative γ of the curve  on  2 has a component perpendicular to  2 and a component tangent to  2 ; that is, where the dot "⋅" denotes the derivative with respect to the parameter of the curve.The norms ‖ γ ⊺ ‖ and ‖ γ ⊥ ‖ are called the geodesic curvature and the normal curvature of  on  2 , respectively.The curve  is called geodesic (resp., asymptotic line) if and only if its geodesic curvature   (resp., normal curvature   ) vanishes.
In our framework, the following results provide some characterizations for the parameter curves of the Hasimoto surfaces to be geodesics and asymptotic lines in G 1  3 .
Theorem 7. Let  2 be a Hasimoto surface given by (23) From ( 21), we derive that the geodesic curvature   of the parameter curves vanishes.This gives first statement.

1 3
Let   and   be the open intervals of R and =   ×   open domain of R 2 .A Hasimoto surface  2 is the surface traced out by a curve in G 1 3 as it evolves over time according to this evolution equation: in G 1 3 .Then (i) the -parameter curves on  2 are geodesics of  2 , (ii) the -parameter curves on  2 are geodesics of  2 if and only if ()  +    vanishes, where  and  are, respectively, the curvature and the torsion of the curve (, ) for ∀ ∈   .Proof.Let us assume that  2 is a Hasimoto surface given by (23) in G 1 3 .Then the geodesic curvature of the -parameter curves on  2 is the tangential component of r  ; that is, = S ⋅ r  = 1 √      [( 2 )  ] 2 − [( 3 )  ] 2     (r  ⋅ r  ) .