Natural and Conjugate Mates of Frenet Curves in Three-Dimensional Lie Group

In this study, we introduce the natural mate and conjugate mate of a Frenet curve in a three dimensional Lie group $ \mathbb{G} $ with bi-invariant metric. Also, we give some relationships between a Frenet curve and its natural mate or its conjugate mate in $ \mathbb{G} $. Especially, we obtain some results for the natural mate and the conjugate mate of a Frenet curve in $ \mathbb{G} $ when the Frenet curve is a general helix, a slant helix, a spherical curve, a rectifying curve, a Salkowski (constant curvature and non-constant torsion), anti-Salkowski (non-constant curvature and constant torsion), Bertrand curve. Finally, we give nice graphics with numeric solution in Euclidean 3-space as a commutative Lie group.


Introduction
In the theory of curves in differential geometry, "generating a new curve from a regular curve, examining the relationships between them and obtaining new characterizations for them" is always a matter of curiosity and has taken its place among popular topics. In this sense, in the category of curves associated with the Frenet vector fields of regular curves; Bertrand curve, involute-evolute curve, Mannheim curve, principal-direction or binormal-direction curve, and also with respect to their position vector; rectifying curve, osculating curve, normal curve are among the leading examples. These curves and their geometric properties have been studied by many authors in different ambient spaces.
From the fundamental theorem of curves, we have general information about the structure of the curve if the curvatures of any regular curve are known. Therefore, in the theory of curves, "to give the characterization of the curve in terms of curvatures" is another attracted topic. For example; in Euclidean 3-space, for any regular curve with curvature κ and torsion τ , the following characterizations are well known: • κ = 0 iff it is a straight line, • τ = 0 iff it is a planar curve, • τ = 0 and κ = c is a non-zero constant iff it is a circle with radius c −1 [2], and under assuming that κ = 0; • (1/κ) (1/τ ) 2 + (1/κ) 2 = r 2 iff it is a spherical curve which is lying on a sphere with radius r [2], • the ratio (τ /κ) is a constant iff it is a general helix. Especially, both of the curvatures are a non-zero constant iff it is a circular helix [12,20], • the ratio (τ /κ) is a linear function with respect to arc-length parameter iff it is rectifying curve [3], • κ 2 / τ 2 + κ 2 3/2 (τ /κ) is a constant iff it is a slant helix [10], • κ is a constant but τ is a non-constant function iff it is a Salkowski curve. Conversely, κ is a nonconstant function but τ is a constant iff it is a anti-Salkowski curve [13,19].
The other versions of the associated curves and the special curves were also defined in various ambient spaces such as Riemannian or Lorentzian space forms. In particularly, these curves and their geometric properties are also studied in three-dimensional Lie groups. In this sense, general helix [5], slant helix [15], rectifying curve [1] and recently Darboux helix [17] as special curves; Bertrand curve [16], Mannheim curve [9] as mate curves; and also principal-direction curve [11] as associated curves, are studied in a three-dimensional

Preliminary
Let G be a three dimensional Lie Group with a bi-invariant Riemannian metric <, > and g be the Lie algebra of G, which is consisted of all smooth vector fields of G. Then g is isomorphic to T e G, where e is identity of G. Moreover the following equations and are satisfied with respect to bi-invariant metric for all X, Y, Z ∈ g, where ∇ is the Levi-Civita connection of Lie group G.
Let γ : I ⊂ R → G be arc-lenghted (unit speed) curve and {X 1 , X 2 , X 3 } be an orthonormal basis of g.
j=1 v j X j along the curve γ where u i and v j are smooth functions from I to R. Moreover, Lie bracket of two vector fields U and V along the curve γ is given by Let ∇ T U is the covariant derivative of U along the curve γ and it is given by left-invariant vector field to γ, then U = 0.
Let γ : I ⊂ R → G be a unit speed curve with the Frenet-Serret apparatus {T, N, B, κ, τ } such that κ > 0. Then the Frenet-Serret equations of γ is given by where the curvature and torsion of γ is If torsion τ = 0, the curve γ is called a Frenet curve in G. Also, a smooth function τ G which is called Lie group torsion of γ, is given by  are satisfied [5,21].
Let τ G be the Lie group torsion of γ, which is given by (2.7). Then, we easily see that where τ G is the Lie group torsion of γ [15].
Theorem 2.1. The curve γ is a general helix in G iff its harmonic function is a constant function [5,15].
Theorem 2.2. The curve γ is a slant helix in G iff the function is a constant function, where H is harmonic curvature of γ [15].
which is satisfying the following equations where "×" is (natural) cross product in three-dimensional Lie algebra g. Moreover, we define the vector field is given by which is satisfying the following equations Hence Ω is called extrinsic Darboux vector of the curve γ with respect to usual derivative and its length is The extrinsic co-Darboux vector of the curve γ is given by It is easily seen that Ω * corresponds to usual derivative ( ) of the principal normal of the curve γ. That is, Definition 2.4. Let γ : I ⊂ R → G be an arc length parametrized curve, then a curve α : Remark 2.1. A left shift is a canonical map from the tangent space of G to the Lie algebra g [8].
Proposition 2.2. Let γ : I ⊂ R → G be a curve, then there exists a left shift (unique up to initial conditions) .
Definition 2.5. The curve γ is called a spherical curve in G if the left shift α of γ lies on the unit central sphere in g (i.e. α(t), α(t) = 1 for all t ∈ I) [5].
in G.
Theorem 2.4. Let γ be a unit speed Frenet curve in G with Frenet-Serret apparatus {T, N, B, κ, τ }. Then γ is a spherical curve whose left shift is lying a sphere with radius r if and only if the following equations satisfy: where H is harmonic curvature of γ.
Remark 2.2. Under special cases, it is known that a three dimensional Lie group G with bi-invariant as a commutative group, and also 3-dimensional unit sphere S 3 (or special unitary group SU(2)), 3-dimensional special orthogonal group SO(3). Especially, G is a commutative group, S 3 or SO(3) when τ G = 0, 1 or 1 2 , respectively [5,7]. Thus, we say that a three dimensional Lie group G with bi-invariant metric has a very rich structure and also, the following results are extended versions of the results which is given in [6].

Natural mates of Frenet curves in G
In this section, we introduce natural mate of a Frenet curve in three dimensional Lie group G with biinvariant metric. Also, we give some relationship between Frenet curve and its natural mates. Moreover, we obtain some results for the natural mate of a Frenet curve which is especially a general helix, a slant helix, a spherical curve or a curve with constant curvature. Remark 3.1. It is easily seen that the natural mate curve of the Frenet curve γ is given by integral of principal normal N from Definition 3.1. So, natural mate curve is same from algebraic viewpoint with principal-direction curve in [11]. But natural mate curve is different from geometric viewpoint since it is defined as along the Frenet curve γ in three-dimensional Lie group G. Now, let γ : I ⊂ R → G be a unit speed Frenet curve in G with the Frenet-Serret apparatus {T, N, B, κ, τ }.
Then, it is easily seen that N, Ω * ω , Ω ω is an orthonormal basis in Lie algebra g of G along the curve γ, where Ω, ω, Ω * are defined by (2.14), (2.16), (2.17), respectively. Now, by using (2.10), (2.11), (2.16), we get Also, by using equations (2.9) with these functions, we have Moreover, equations (3.1) means that there exists a unit speed Frenet curve β in G with Frenet-Serret apparatus {T , N , B, κ, τ } which is satisfying the following equations . Thus, we have T = N along the curve γ by equations (3.2). That is, the curve β is natural mate of the curve γ by Definition 3.1. Then, we obtain the following theorem for natural mate curve of a Frenet curve in G.   Remark 3.2. We remark that Theorem 3.1, Corollary 3.1 and Corollary 3.2 are same from algebraic viewpoint with Theorem 5, Theorem 7 and Theorem 8 in [11], respectively. Now, we give the following characterization for natural mate β of the curve γ which is a rectifying curve.
Proof. We suppose that γ is rectifying curve in G with arc-length parameter s. Then, by Theorem 2.3, the harmonic curvature of γ is given by H(s) = as + b where a = 0 and b are constants. Now, let the curvature and the torsion of the natural mate of γ be κ and τ , respectively. Then, by the equations (3.2), Hence, we get easily the equation (3.3) from the last equations.
On the other hand, let the curvatures of γ and its natural mate satisfy the the equation (3.3). Then, by using the equations (3.2), we obtain that H (s) = a where a is non-zero constant and s is arc-length parameter of γ. This means that H (s) = as + b and so γ is a rectifying curve in G by Theorem 2.3. Now, by using Theorem 3.1, we give a result for natural mate β of γ which is especially a spherical curve.
Corollary 3.4. Let γ be a unit speed Frenet curve in G with curvature κ and torsion τ . Then γ is a spherical curve whose left shift is lying a sphere with radius r in Lie algebra g of G if and only if the curvature κ and torsion τ of its natural mate satisfȳ Proof. We assume that γ is a spherical curve whose left shift is lying a sphere with radius r in Lie algebra g of G with curvature κ and torsion τ . Then, In that case, the equation (3.7) gives Then, after differentiation of the equation (3.8) and bearing the equation (3.7) in mind, we obtain Thus, the last equation means that γ is a spherical curve in G by the equation (2.19).

Spherical natural mates in G
Theorem 4.1. Let γ be a Frenet curve in G with constant curvature κ = c > 0, then its natural mate β is a spherical curve whose left shift is lying a sphere with radius (1/c) in g. The converse holds if the torsion of the natural mate β is not equal to the Lie group torsion of β, that is τ = τ G .
Proof. We suppose that γ is a Frenet curve in G with constant curvature κ = c > 0. Then, by using the the equations (3.2), the curvatures of its natural mate β are given bȳ where the harmonic curvature function of γ is H = (τ − τ G ) /c.
Hence, by Theorem 2.4, the natural mate β is a spherical curve whose left shift is lying a sphere with radius (1/c) in g.
Case (2): If τ = τ G , then by using the equation (4.1), we get 1 κ where the harmonic curvature function of the natural mate β is H = (τ − τ G ) /κ. Hence, by Theorem 2.4, the natural mate β is a spherical curve in G.
Conversely, under the case τ = τ G , we assume that the natural mate β is a spherical curve whose left shift is lying a sphere with radius (1/c) in g. Then, by Theorem 2.4, we have and so, we getκ If we integrate the last equation after putκ = c u, then we have and so,κ = c sec (τ − τ G ) ds .

(5.2)
Proof. We suppose that the curvature κ of the natural mate β is a constant c > 0. Then, by the equations such that κ = ω. Also, we get Now, we give the following characterization for a spherical curve γ in G whose the curvature of its natural mate β is a non-zero constant.
Theorem 5.2. Let γ be a Frenet curve in G with arc-length parameter s, and β be its natural mate whose the curvature κ is a non-zero constant c. Then γ is a spherical curve in G if and only if there is a positive constant a ≥ c such that the torsion τ of β satisfies where H is the harmonic curvature function of γ. Moreover, by the equation (5.5), we get and so, Now, by putting the equations (5.5) and (5.8) in the equation (5.6), we obtain By differentiating of the last equation and bearing the equations (5.6) and (5.7) in mind, we get Consequently, we obtain the equation (2.19). That is, γ is a spherical curve in G.
Now, we give the following result which is obtained by Corollary 3.4 for a spherical curve in G whose natural mate has non-zero constant curvature.
Corollary 5.1. Let γ be a Frenet curve in G with curvature κ and torsion τ , and also β be its natural mate with non-zero constant curvature κ = c > 0 and torsion τ . Then γ is a spherical curve whose left shift is lying a sphere with radius (1/r) in g if and only if τ − τ G = 0 or τ − τ G = ∓κ √ r 2 κ 2 − 1.
Proof. We assume that γ is a spherical curve whose left shift is lying a sphere with radius (1/r) in g. Then, by Corollary 3.4, the equation (3.4) is satisfied as a necessary and sufficient condition. Also, since the natural mate β of γ has non-zero constant curvature κ = c > 0, we obtain Under the condition τ − τ G = 0, the equations (6.1) means that there exists a unit speed Frenet curve γ * in G with Frenet-Serret apparatus {T * , N * , B * , κ * , τ * } which is satisfying the following equations where τ G * = 1 2 [T * , N * ], B * . Thus, we have T * = B along the curve γ by the equations (6.2). That is, the curve γ * is the conjugate mate of the curve γ by Definition 6.1. Moreover, and so, Thus, we obtain the following theorem for conjugate mate of a Frenet curve in G. By using Theorem 6.1, we get easily following results for conjugate mate γ * of a Frenet curve γ which is a general helix or slant helix. Proof. Let the harmonic curvatures of γ and γ * be H and H * , respectively. Then, by using (6.2), we have Thus, the proof is clear.
Hence, the proof is clear. Now, we give the following results by Theorem 3.1 and Theorem 6.1 for involute-evolute curves (see [15]) and Bertrand curve couple (see [16]) . Corollary 6.3. Let γ : I ⊂ R → G be a unit speed Frenet curve with τ − τ G = 0. Then there exists a unique pair of a unit speed curve β : I ⊂ R → G and a Frenet curve γ * : I ⊂ R → G, such that the curves γ, β and γ * are mutually orthogonal curves (i.e. involute-evolute curves). Thus, we conclude that the curve β and the curve γ * are congruent from the equations (6.5), (6.6) and (6.7).
Also, τ − τ G = c is non-zero constant from hypothesis, so the curvature κ * of the Frenet curve γ * is non-zero constant |c| > 0 by using (6.2). Finally, if we apply Theorem 4.1 and take into account that the curves β and γ * are congruent, the proof is completed.
Finally, when G is Euclidean 3-space as a commutative group (i.e. τ G = 0), we give graphics of some special curves jointly with their natural mate and conjugate mate. In addition, we should also mentioned that the graphics are obtained by method of numerical solution in Mathematica.