SIMILAR AND SELF-SIMILAR NULL CARTAN CURVES IN MINKOWSKI-LORENTZIAN SPACES

In this paper, differential invariants of null Cartan curves are studied in (n+2) dimensional Lorentzian similarity geometry. The fundamental theorem for a null Cartan curve in similarity geometry is investigated and the characterization of all self-similar null Cartan curves parameterized by de Sitter parameter in Minkowski space-time is given.


Introduction
A similarity transformation of Euclidean space, which consists of a rotation, a translation and an isotropic scaling, is an automorphism preserving the angles and ratios between lengths. The structure consisting of unchanging geometric properties under the similarity transformation is called similarity geometry. The whole Euclidean geometry can be considered as a class of similarity geometry. The similarity transformations are studied in most areas of the pure and applied mathematics. For example, S. Li [23] presented a system for matching and pose estimation of 3D space curves under the similarity transformation. Brook et al. [5] discussed various problems of image processing and analysis by using the similarity transformation. Sahbi [26] investigated a method for shape description based on kernel principal component analysis (KPCA) in the similarity invariance of KPCA. On the other hand, the self-similar objects, whose images under the similarity map are themselves, have had a wide range of applications in areas such as fractal geometry, dynamical systems, computer networks, statistical physics and so on. The Cantor set, the von Koch snowflake curve and the Sierpinski gasket are some of most famous examples of such objects (see [14,21]). Recently, the self-similarity started playing a role in algebra as well, first of all in group theory [17].

H. ŞIMŞEK
Bonnor [4] introduced the Cartan frame to study the behaviors of a null curve and proved the fundamental existence and congruence theorems in Minkowski spacetime. Bejancu [2] represented a method for the general study of the geometry of null curves in Lorentz manifolds and, more generally, in semi-Riemannian manifolds (see also the book [11]). Ferrandez, Gimenez and Lucas [15] gave a reference along a null curve in an n-dimensional Lorentzian space. They showed the fundamental existence and uniqueness theorems and described the null helices in higher dimensions. Cöken and Ciftci [10] studied null curves in the Minkowski space-time and characterized pseudo-spherical null curves and Bertrand null curves.
The study of the geometry of null curves has a growing importance in the mathematical physics. The null curves are useful to find the solution of some equations in the classical relativistic string theory (see [6,19,20]) Moreover, there exists a geometric particle model associated with the geometry of null curves in the Minkowski space-time (see [16,24]).
Berger [3] represented the broad content of similarity transformations of Euclidean space. Encheva and Georgiev [12,13] studied the differential geometric invariants of curves according to a similarity in the Euclidean n-space. Chou and Qu [9] showed that the motions of curves in two, three and n-dimensional (n > 3) similarity geometries correspond to the Burgers hierarchy, Burgers-mKdV hierarchy and a multi-component generalization of these hierarchies. Şimşek andÖzdemir [27] introduced the geometry of non-lightlike curves in the n-dimensional Lorentzian similarity geometry. Ateş et.al. [1] studied the similarity invariants of Frenet curves by considering the parametrization of any spherical indicatrix curve in Eucliedan space E n . Kamishima [22] examined the properties of compact Lorentzian similarity manifolds using developing maps and holonomy representations. The main idea of this paper is to study the differential geometry of a null curve under the similarity mapping.
The scope of paper is as follows. First, we give basic information about null Cartan curves. Then, we introduce a new parameter, which is called de Sitter parameter that is invariant under the similarity transformation. We represent the differential geometric invariants of a null Cartan curve, which are called shape Cartan curvatures, in (n+2)-dimensional Lorentzian similarity geometry. We prove the uniqueness theorem which states that two null Cartan curves are equivalent according to a similarity mapping. Furthermore, we show the existence theorem that is a process for constructing a null Cartan curve by the shape Cartan curvatures under some initial conditions. Lastly, we obtain the equations of all self-similar null Cartan curves parameterized by the de Sitter parameter in Minkowski spacetime.

Preliminaries
Let u = (u 1 , u 2 , . . . , u n+2 ) , v = (v 1 , v 2 , . . . , v n+2 ) be two arbitrary vectors in Minkowski-Lorentzian space M n+2 . The Lorentzian inner product of u and v can be stated as u · v = uI * v T where I * = diag(−1, 1, . . . , 1). We say that a vector u in M n+2 is called spacelike, null (lightlike) or timelike if u · u > 0, u · u = 0 or u · u < 0, respectively. The norm of the vector u is defined by ∥u∥ = |u · u|. The pseudohyperbolic space (or anti-de Sitter space) is defined by and pseudo-sphere (or de Sitter space) is defined by A basis B = {L, N, W 1 , . . . , W n } is said pseudo-orthonormal if it satisfies the following conditions: Now, we consider the mapping A : L ,N,W 1 , · · · ,W n → (L, N, W 1 , · · · , W n ) of one pseudo-orthonormal basis onto another at any point P in M n+2 , which is given as either λε n−1 cos θ + λε n sin θ 0 0 0 0 · · · 0 cos θ − sin θ λε n−1 sin θ − λε n cos θ 0 when n is even, or the same matrix with the additional row (column) 0 0 · · · 1 when n is odd, where λ, ε i (1 ≤ i ≤ n) and θ are real constants and λ ̸ = 0. The image of pseudo-orthonormal basis under the mapping A is a pseudo-orthonormal basis. Moreover, we have A T J * A = J * , det A = 1 where and the orientation is preserved by (1). Bonnor [4] defined the mapping A as a null rotation. A null rotation at P is equivalent to a Lorentzian transformation between two sets of natural coordinate functions whose values coincide at P . A curve locally parameterized by γ : J ⊂ R → M n+2 is called a null curve if d dt γ(t) ̸ = 0 is a null vector for all t. We know that a null curve γ(t) satisfies [11]). If the acceleration vector of the null curve is a unit vector, that is, d 2 dt 2 γ(t) · d 2 dt 2 γ(t) = 1, then, null curve γ(t) in M n+2 is said to be parameterized by pseudo-arc. If the acceleration vector of the null curve is not a unit vector, then the pseudo-arc parametrization becomes as the following There exists a unique Cartan frame C γ := {L, N, W 1 , · · · , W n } of the Cartan curve parameterized by a pseudo arc-parameter s such that the following equations are satisfied where N is a null vector called null transversal vector field, and C γ is pseudoorthonormal, γ ′ , γ ′′ , . . . , γ (i+2) and {L, N, W 1 , . . . , W i } have the same orientation for 2 ≤ i ≤ n − 1, C γ is positively oriented and the differentiation with respect to s is denoted by prime " ′ " . The functions κ, τ , κ j (3 ≤ j ≤ n) are called the Cartan curvatures of γ (s) and are given as for the pseudo-arc parameter s, where D j denotes the j-th order main determinant of the matrix of the metric with respect to γ ′ , γ ′′ , . . . , γ (n+2) . We know that τ < 0, , and κ n > 0 or κ n < 0 according to γ ′ , γ ′′ , . . . , γ (n+2) is positively or negatively oriented, respectively. More information about the geometry of null curves can be found in the papers [2], [4], [11] and [15].

Geometric Invariants of Null Curves Under a Similarity Map
In this section, we introduce the similarity geometry of null curves in M n+2 . A null-similarity (n-similarity) f : M n+2 → M n+2 is determined by where µ > 0 is a real constant, A is a null rotation and C is a translation vector. The n-similarity transformations form a group under the composition of maps and denoted by Simn M n+2 . The n-similarity transformations in M n+2 preserve the orientation. Let γ (t) : J ⊂ R → M n+2 be a null curve. The image of γ under f ∈ Simn M n+2 is denoted by β. Then, the null curve β can be stated as The pseudo-arc length function β starting at t 0 ∈ J is where s ∈ I ⊂ R is pseudo-arc parameter of γ : I → M n+2 . We can compute the Cartan curvatures κ β √ µs and τ β √ µs of β by using (4) as We define W 1 -indicatrix γ W1 of the null curve γ parameterized by γ W1 (s) = W 1 (s). The W 1 -indicatrix is a pseudo-spherical non-null curve lies on the de Sitter (n+1)-space S n+1 1 (1). If we state the arc-length parameter of γ W1 as σ γ , we can find dσ γ = 2 |κ γ |ds. The arc-length element dσ γ is invariant under the n-similarity transformation since the equality dσ β = dσ γ can be easily found, where σ β is the de Sitter parameter of β. The parameter σ γ is called de Sitter parameter of γ. Therefore, we reparametrize a null curve by the de Sitter parameter so that we can study the differential geometry of a null curve under the n-similarity transformation.
The derivative formulas of γ and C γ with respect to σ γ are given by and Similarly, we can find the same formulas (9) and (10) for the null curve β. Now, we construct a new frame corresponding to n-similarity transformation for a null curve. Let's define the functions which are invariant under the n-similarity since we get the equalities If we set L sim = 2 |κ γ |L, then we get a unit spacelike vector such that L sim · W sim 1 = 0. From [2], we know that there exists a null vector N sim satisfying L sim · N sim = 1, N sim · W sim 1 = 0, in the space spanned by {γ ′ , γ ′′ , γ ′′′ } such that N sim can be given in the form which satisfies the relations N sim ·N sim = 0, L sim ·N sim = 1 and N sim ·W sim where Let's consider the pseudo-orthogonal frame Since, from (5) , we can obtain f (H γ i ) = H β i , i = 1, · · · n + 2, the pseudoorthogonal frame C H γ is invariant according to n-similarity map. Then, using (9) and (11) , we get the derivative formulas of C H γ as the following We can consider the equation (13) as the Frenet-Serret equation of a null Cartan curve γ according to the pseudo-orthogonal moving frame C H γ under the group Simn M n+2 . As a result, the following theorem is obtained.

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H. ŞIMŞEK Theorem 1. Let γ : I → M n+2 be a null Cartan curve with pseudo-de Sitter parameter σ γ and {κ γ , τ γ ,κ iγ (3 ≤ i ≤ n)} be Cartan curvatures of γ with the Cartan frame C γ . Then, the functions and the pseudo-orthogonal frame C H γ are invariant under the n-similarity transformation in M n+2 and the derivative formulas of C H γ with respect to σ γ are given by the equation (13) . Definition 1. The functionsξ γτ γ andκ iγ (3 ≤ i ≤ n) are called shape Cartan curvatures of a null Cartan curve γ and the pseudo-orthonormal frame C sim γ are called shape Cartan frame of γ.

The Fundamental Theorem for a Null Curve
The existence and uniqueness theorems were shown by [2], [15] and [4] for a null Cartan curve under the Lorentz transformations. This notion can be extended with respect to Simn M n+2 for the null Cartan curves parameterized by de Sitter parameter.
Proof. Let κ γ , τ γ , κ iγ and κ β , τ β , κ iβ (3 ≤ i ≤ n) be the Cartan curvatures and also s and s * be the pseudo-arc length parameters of γ and β, respectively. Using the equalityκ γ =κ β , we get |κ γ | = µ |κ β | for some real constant µ > 0. Then, the equalityτ γ =τ β imply τ γ = µτ β . Therefore, we find ds = 1 √ µ ds * from the definition of de Sitter parameter σ. There exists a Lorentzian motion φ = A • T of M n+2 satisfying the equality φ (γ (σ 0 )) = β (σ 0 ) for any fixed σ 0 ∈ I, where A is a null rotation and T is a translation map, such that φ maps the pseudo-orthonormal frame C sim γ to pseudoorthonormal frame C sim β . Therefore, the map g = µφ : M n+2 → M n+2 is a nsimilarity transformation of M n+2 . Let's define a function Φ : I → R as the following Taking derivative of this function with respect to σ, we conclude that Then, we obtain the following equation from (9). Also, we can find The following theorem shows that every two functions determine a null Cartan curve according to a n-similarity under some initial conditions. Theorem 3. Let z i : I → R, i = 1, 2, . . . n be smooth functions and L 0sim , N 0sim , W 0sim 1 , W 0sim 2 , . . . , W 0sim n be a pseudo-orthonormal frame at a point x 0 in the Minkowski space M n+2 . According to a n-similarity, there exists a unique null Cartan curve γ : I → M n+2 parameterized by the de Sitter parameter σ such that γ satisfies the following conditions: (i) There exists σ 0 ∈ I such that γ (σ 0 ) = x 0 and the shape Cartan frame of γ at Proof. Let us consider the following system of differential equations with respect to a matrix-valued function 88 H. ŞIMŞEK with a given matrix The system (15) has a unique solution K (σ) which satisfies the initial conditions . Also, we have where I is the unit matrix since L 0sim , N 0sim , W 0sim 1 , W 0sim 2 , . . . , W 0sim n is the pseudo-orthonormal (n+2)-frame. As a result, we find J * X T (σ) J * X (σ) = I for all σ ∈ I. This means that the vector fields L sim , N sim , W sim 1 , W sim 2 , . . . , W sim n form a pseudo-orthonormal frame field in M n+2 . Let γ : I → M n+2 be a null curve given by By the equality (15), we get that γ (σ) is a similar null Cartan curve with the curvaturesκ γ (σ) = z 1 (σ),τ γ (σ) = z 2 (σ) andκ iγ (σ) = z i (σ) (3 ≤ i ≤ n) in M n+2 .

Self-similar Null Cartan Curves
In this section, the concept of self-similarity is applied to null Cartan curves. Self-similar spacelike and timelike curves were studied by [27] and were defined as curves with constant p-shape curvatures. This idea can be extended to a null Cartan curve γ : I → M n+2 ; that is, γ is called self-similar if shape Cartan curvatures of γ are constant.
A null curve is called a null helix if it has the constant Cartan curvatures which are not all zero in M n+2 . A null helix is automotically a self-similar null Cartan curve in M n+2 . Thus, null helices can be considered as a subclass of self-similar null Cartan curves. 5.1. Self-similar null Cartan curves in M 4 . Now, we determine the parametrizations of all self-similar null Cartan curves by means of the constant shape Cartan curvatures in the Minkowski space-time. They can be examined by separating into four different cases as follows. For each case, we choose the initial conditions (18) in the example 1.
GCase 1: Using the equation (15) , we obtain the following differential equation and by solving this equation, we conclude that if c > 0 If the Case 1 is valid, we use the equation (17) so that we get the following parametrization of the self-similar null Cartan curve when c > 0 and when c < 0. Sinceξ γ 1 = c = ± 1 2 for the Case 1, we obtain for c = 1/2 and If the Case 3 is valid, we use the equation (16) so that we get the following parametrization of the self-similar null Cartan curve when c > 0 and GCase 2: Using the equation (15) , we obtain the following differential equation and by solving this equation, we conclude that If the Case 2 is valid, we use the equation (17) so that we get the following parametrization of the self-similar null Cartan curve If the Case 4 is valid, we use the equation (16) so that we get the following parametrization of the self-similar null Cartan curve γ 6 (σ) = 1 4 √ 2 ( cosh (n 1 σ) + sinh (n 1 σ) n 1 + cosh (n 2 σ) + sinh (n 2 σ) n 2 , cosh (n 1 σ) + sinh (n 1 σ) n 1 − cosh (n 2 σ) + sinh (n 2 σ) n 2 , 4be 2bσ cos (q 2 σ) + 2q 2 e 2bσ sin (q 2 σ) 4b 2 + q 2 2 , −2q 2 e 2bσ cos (q 2 σ) + 4be 2bσ sin (q 2 σ) where n 1 = 2b + q 1 ̸ = 0, n 2 = 2b − q 1 ̸ = 0. From the above calculations, we obtain the following result.  In [4], null helices were defined and found their parametrizations in M 4 . When we compare the parametrizations of null helices with self-similar null curves, we conclude that null helices are a special class of self-similar null Cartan curves in where v = √ κ 2 + τ 2 − κ and r = √ κ 2 + τ 2 + κ and this curve is a kind of self-similar null Cartan curve γ 5 (see also [10] for null helices).
In [10], Theorem 3.2 says that a null Cartan curve γ lies on S 3 1 (r) iff τ γ ̸ = 0 is a constant in M 4 . Then, we conclude that the self-similar null Cartan curve lying on S 3 1 (r) is similar to γ 5 because of the definitionsκ γ andτ γ . On the other hand, in [7], Theorem 3.10 states that there are no null curves lying on H 3 0 (r) in M 4 , which means that there is no a self-similar (similar) null Cartan curve lying on H 3 0 (r) .

Concluding Remarks
In the current paper, the similarity geometry of a null Cartan curve in Minkowski-Lorentzian spaces was investigated and self-similar null Cartan curves were studied in Minkowski space-time. Next study will be about self-similar null Cartan curves in Lorentzian space forms like null helices studied in [15].
The motions of curves in E 2 , E 3 and E n (n > 3) yield the mKdV hierarchy, Schrödinger hierarchy and a multi-component generalization of mKdV-Schrödinger hierarchies, respectively. KS. Chou and C. Qu [9] showed that the motions of curves in two-, three-and n-dimensional (n > 3) similarity geometries correspond to the Burgers hierarchy, Burgers-mKdV hierarchy and a multi-component generalization of these hierarchies by using the similarity invariants of curves in comparison with its invariants under the Euclidean motion. Also, they [8] found that many 1+1dimensional integrable equations like KdV, Burgers, Sawada-Kotera, Harry-Dym hierarchies and Camassa-Holm equations arise from motions of plane curves in centro-affine, similarity, affine and fully affine geometries. The motion of curves on two-dimensional surfaces in E 3 1 was considered by Gürses [18]. Therefore, the motion of Lorentzian similar (null and nonnull) curves in Lorentzian-Minkowski similarity geometries will be investigated as well.

Declaration of Competing Interests
The author declares that he has no competing interest.