A characterization of ccr-curves in R m

We study the curve in Rm for which the ratios between two consecutive curvatures are constant (ccr-curves). We show that closed ccr-curves in Euclidean space Rm are of finite type. We also consider Frenet curves with constant harmonic curvatures and show that an immersed curve in R2n+1 with constant harmonic curvatures Hi at point γ(s0) has a Darboux vertex at that point.


INTRODUCTION
Let γ = γ(s) : I → R m be a regular curve in R m (i.e.γ is nowhere zero), where I is an interval in R. The curve γ is called a Frenet curve of rank r (r ∈ N 0 ) if γ (t), γ (t), ..., γ (r) (t) are linearly independent and γ (t), γ (t), ..., γ (r+1) (t) are no longer linearly independent for all t in I.In this case, Im(γ) lies in an r-dimensional Euclidean subspace of R m .For each Frenet curve of rank r there occur an associated orthonormal r-frame {E 1 , E 2 , ..., E r } along γ, the Frenet r-frame, and r − 1 functions κ 1 , κ 2 , ..., κ r−1 :I −→ R, and the Frenet curvatures, such that , where v is the speed of the curve.
The notion of a generalized helix in R 3 , a curve making a constant angle with a fixed direction, can be generalized to higher dimensions in many ways.In [14], the same definition is proposed in R m .In [9], the definition is more restrictive: the fixed direction makes a constant angle with all the vectors of the Frenet frame.It is easy to check that the definition only works in the odd dimensional case.Moreover, in the same reference, it is proven that the definition is equivalent to the fact that the ratios κ 2 κ 1 , κ 4 κ 3 , ..., κ i being the curvatures, are constant.
In [15] Uribe-Vargas proved that the immersed curve in R 2n+1 , n 1 has a Darboux vertex at point γ(s 0 ) if and only if [11] has considered the Frenet curves in R m which have constant curvature ratios (i.e., κ 2 κ 1 , κ 3 κ 2 , κ 4 κ 3 ... are constant).The Frenet curves with constant curvature ratios are called ccr-curves.In the present study we prove that if the harmonic curvatures H i of the immersed curve in R 2n+1 are constant at point γ(s 0 ), then γ has a Darboux vertex at that point.
We also prove that every closed ccr-curve is of finite type.

W-CURVES
Definition 1.A Frenet curve of rank r for which κ 1 , κ 2 , ..., κ r−1 are constant is called (generalized) screw line or helix [6].Since these curves are trajectories of the 1-parameter group of the Euclidean transformations, Klein and Lie [10] called them W-curves.
A unit speed W-curve of rank 2k in R m has the parameterization of the form and a unit speed W-curve of rank (2k + 1) has the parameterization of the form where a 0 , b 0 , a 1 , ..., a k , b 1 , ..., b k are constant vectors in R m and µ 1 < µ 2 < ... < µ k are positive real numbers.So, a W-curve of rank 1 is a straight line, a W-curve of rank 2 is a circle, and a W-curve of rank 3 is a right circular helix.
The subset of R 2n parameterized by where By analogy, the subset of R 2n+1 parameterized by where u i ∈ R and a is a real constant, will be called a flat torus in R 2n+1 .We give the following examples.
Example 1.Any curve in a flat torus of the kind has all its curvatures constant (i.e.W-curve).These curves are the geodesics of the flat tori and it is proven in [13] that they are twisted curves if and only if the constants m i = m j for all i = j.For closed twisted curves see also [13].
Example 2. (Helices in S 3 ) Let S 3 be the unit 3-sphere imbedded in the Euclidean 4-space E 4 .A model helix in S 3 ⊂ E 4 is given by γ(s) = (cos φ cos(as), cos φ sin(as), sin φ cos(bs), sin φ sin(bs)), Here s is the arclength parameter.It is easy to see that γ lies in the flat torus: Example 3. The Frenet curve α : I → R 4 given by the parameterization is a spherical W-curve (with radius 1), (see [11]) where,

CURVES OF FINITE TYPE
Let f (s) be a periodic continuous function with period 2πr.Then it is well known that f (s) has a Fourier series expansion given by where a k and b k are the Fourier coefficients defined by Let γ be a closed curve of length 2πr.If x : γ → R m is an isometric immersion, then If x is of finite type, each coordinate function x i satisfies the following homogeneous ordinary differential equation with constant coefficients: for some integer k ≥ 1 and constant c 1 , ..., c k .Because our solutions x i of the above differential equation are periodic solutions with period 2πr, each x i is a finite linear combination of the following particular solutions: Therefore, each x i is of the form for some suitable constant c i , a A (t), b A (t) (A = 1, ..., n) and integers p A , q A .Thus each x i has a Fourier series expansion of finite sum.Similarly, if each x i has a Fourier series expansion of finite sum, then x is of finite type (see [4,5,7]).Thus, using the above theorems, we have the following corollary.
Corollary 2. Every closed k-type curve γ in R m can be written in the form where T (x) = {t 1 ,t 2 , ...t k } is the order of the curve and a 0 , a 1 , ..., a k , b 1 , ..., b k are vectors in R m such that for any i in {1, 2, ..., k}, a i and b i are not simultaneously zero.Moreover, if q = t k is the upper order of γ, then a q = b q = 0.
Corollary 3. Every null k-type curve γ in R m can be written in the form where a 0 , b 0 , a 1 , ..., a k , b 1 , ..., b k are vectors in R m such that b 0 = 0 and for any i in {1, 2, ..., k}, a i and b i are not simultaneously zero.Moreover, a q = b q = 0, where q is the upper order of the curve γ.

Corollary 4. [2]
1) Every k-type curve of R m lies in an affine 2k-subspace R 2k of R m .
2) Every null k-type curve of R m lies in an affine (2k − 1)-subspace R 2k−1 of R m .

GENERALIZED HELICES
In the present section we give some well-known definitions of harmonic curvature and Darboux vertex of a curve in R m .We prove that the immersed curve in R m with constant harmonic curvatures H i at point γ(s 0 ) has a Darboux vertex at that point.
Definition 2. Let γ : I ⊂ R → R m be a regular curve of rank r with unit speed.For 2 ≤ j ≤ r − 2, the functions H j : I → R defined by are called the harmonic curvatures of γ, where κ 1 , κ 2 , ..., κ r−1 are Frenet curvatures of γ which are not necessarily constant and ∇ is the Levi-Civita connection [12].For more details see also [1].
By the use of ( 3) and ( 4) we get the following result.
By Proposition 5 and Theorem 7 we get the following results.
Corollary 8. Let γ : I ⊂ R → R 2n+1 be a regular curve of rank 2n with unit speed.If the harmonic curvatures H i are constant at the point γ(s 0 ), then γ has a Darboux vertex at that point.

CURVES WITH CONSTANT CURVATURE RATIOS
A curve γ = γ(s) : I → R m is said to have constant curvature ratios (ccr-curve) if all the quotients κ i+1 κ i are constant [11].
As is well known, generalized helices in R 3 are characterized by the fact that the quotient τ κ is constant (Lancret's theorem).It is in this sense that ccr-curves are generalization to R m of generalized helices in R 3 .
In [9] a generalized helix in the m-dimensional space (m odd) is defined as a curve satisfying that the ratios κ 2 κ 1 , κ 4 κ 3 , ... are constant.It is also proven that the curve is a generalized helix if and only if there exists a fixed direction which makes constant angles with all the vectors of the Frenet frame.
Obviously, ccr-curves are a subset of generalized helices in the sense of [9].
Corollary 10.Every W-curve is a ccr-curve.
Lemma 11. [11] Let β be a ccr-curve with non-constant curvature.Then Frenet's formulae of β are reduced to a linear system of first order differential equations with constant coefficients for some constants c 2 , ..., c n−1 .

Lemma 12.
[16] Let dx dt = Ax(t) be the linear system of first order differential equations with constant coefficients.Then the homogeneous solutions of the system are given by where u i are the eigenvectors, λ i are the eigenvalues of the constant coefficient matrix of the system, and d i are arbitrary constants.
We prove the following main result.Proof.Let A be the matrix of constant coefficient of system (6).Due to the skewsymmetry of matrix A, it can have no real eigenvalues other than zero.Due to the fact that the determinant of A vanishes only for odd n, we can say that for odd dimensions, 0 is an eigenvalue, whereas for even dimensions, 0 is an eigenvalue only if k n−1 = 0.
From now on, we shall consider that all the curvatures, and all the constants c i are not zero.Therefore, the eigenvalues are all of multiplicity 1.
Let λ l = α l ± iµ l , l = 1, ..., n 2 , with α l , µ l ∈ R be the nonzero eigenvalues of the coefficient matrix A. For n = 2k, from Lemma 12 the general solution of the system for the first vector becomes Similarly for n = 2k + 1, the general solution of the system for the first vector is

Theorem 1 . [ 3 ]
Let γ be a closed curve of length 2πr in R m .Then isometric immersion x : γ → R m is of finite type if and only if the Fourier series expansion of each coordinate function of γ, γ(s) = a 0 +

Theorem 13 .
(Main Result) Every closed ccr-curve is of finite type.

− → e 1
(u) = k ∑ l=1d l u l e α l u cos(µ l u) + f l u l e α l u sin(µ l u) and similarly for n = 2k + 1, the general solution of the system for the first vector is− → e 1 (u) = a 0 + k ∑ l=1 d l u l e α l u cos(µ l u) + f l u l e α l u sin(µ l u),where a 0 , .u 1 , ..., u k are vectors in R m and d l , f l are arbitrary constants.Condition − → e 1 (u) = 1 for all u implies that all the real parts of the eigenvalues are zero.Therefore, for n = 2k, the general solution of the system for the first vector is− → e 1 (u) = k ∑ l=1d l u l cos(µ l u) + f l u l sin(µ l u).