A characterization of ccr-curves in [R.sup.m]/Ruumi [R.sup.m] ccr-koverate iseloomustamine.

1. INTRODUCTION

Let [gamma] = [gamma](s): I [right arrow] [R.sup.m] be a regular curve in [R.sup.m] (i.e. [parallel]g'[parallel] is nowhere zero), where I is an interval in R. The curve [gamma] is called a Frenet curve of rank r (r [member of] [N.sub.0]) if [gamma]'(t), [gamma]"(t), ..., [[gamma].sup.(r)](t) are linearly independent and [gamma]'(t), [gamma]"(t), ..., [[gamma].sup.(r+1)](t) are no longer linearly independent for all t in I. In this case, Im([gamma]) lies in an r-dimensional Euclidean subspace of [R.sup.m]. For each Frenet curve of rank r there occur an associated orthonormal r-frame {[E.sub.1], [E.sub.2], ..., [E.sub.r]} along [gamma], the Frenet r-frame, and r - 1 functions [[kappa].sub.1], [[kappa].sub.2], ..., [[kappa].sub.r-1]: I [right arrow] R, and the Frenet curvatures, such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where v is the speed of the curve.

In fact, to obtain [E.sub.1], [E.sub.2], ..., [E.sub.r] it is sufficient to apply the Gram-Schmidt orthonormalization process to [gamma]'(t), [gamma]"(t), ... [[gamma].sup.(r)](t). Moreover, the functions [[kappa].sub.1],[[kappa].sub.2], ..., [[kappa].sub.r-1] are easily obtained as by-product during this calculation. More precisely, [E.sub.1], [E.sub.2], ..., [E.sub.r] and [[kappa].sub.1],[[kappa].sub.2], ..., [[kappa].sub.r-1] are determined by the following formulas [8]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where k [member of] {2,3, ..., r}. It is natural and convenient to define Frenet curvatures [[kappa].sub.r] = [[kappa].sub.r+1] = ... = [[kappa].sub.m-1] = 0. It is clear that [E.sub.1], [E.sub.2], ..., [E.sub.r] and [[kappa].sub.1],[[kappa].sub.2], ..., [[kappa].sub.r-1] can be defined for any regular curve (not necessarily a Frenet curve) in the neighbourhood of a point s0 for which [gamma]'([s.sub.0]), [gamma]"([s.sub.0]), ..., [[gamma].sup.(r)]([s.sub.0]) are linearly independent.

The notion of a generalized helix in [R.sup.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.sup.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 [[kappa].sup.2]/[[kappa].sub.1], [[kappa].sub.4]/[[kappa].sub.3], ..., [[kappa].sub.i] being the curvatures, are constant.

In [15] Uribe-Vargas proved that the immersed curve in [R.sup.2n+1], n [greater than or equal to] 1 has a Darboux vertex at point [gamma]([s.sub.0]) if and only if ([[kappa].sub.1]/[[kappa].sub.2])' = 0, ([[kappa].sub.3]/[[kappa].sub.4])' = 0, ..., ([[kappa].sub.2k-1]/[[kappa].sub.2k])' = 0.

Recently, Monterde [11] has considered the Frenet curves in [R.sup.m] which have constant curvature ratios (i.e., [[kappa].sub.2]/[[kappa].sub.1], [[kappa].sub.3]/[[kappa].sub.2], [[kappa].sub.4]/[[kappa].sub.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.sub.i] of the immersed curve in [R.sup.2n+1] are constant at point [gamma]([s.sub.0]), then [gamma] has a Darboux vertex at that point.

We also prove that every closed ccr-curve is of finite type.

2. W-CURVES

Definition 1. A Frenet curve of rank r for which [[kappa].sub.1],[[kappa].sub.2], ..., [[kappa].sub.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.sup.m] has the parameterization of the form

[gamma](s) = [a.sub.0] + [k.summation over (i=1)] ([a.sub.i] cos [[mu].sub.i]s + [b.sub.i] sin [[mu].sub.i]s)

and a unit speed W-curve of rank (2k+1) has the parameterization of the form

[gamma](s) = [a.sub.0]+[b.sub.0]s+ [k.summation over (i=1)] ([a.sub.i] cos [[mu].sub.i]s + [b.sub.i] sin [[mu].sub.i]s),

where [a.sub.0],[b.sub.0],[a.sub.1], ..., [a.sub.k], [b.sub.1], ..., [b.sub.k] are constant vectors in [R.sup.m] and [[mu].sub.1] < [[mu].sub.2] < ... < [[mu].sub.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.sup.2n] parameterized by

[??]([u.sub.1], [u.sub.2], ..., [u.sub.n]) = ([r.sub.1] cos([u.sub.1]), [r.sub.1] sin([u.sub.1]), [r.sub.2] cos([u.sub.2]), [r.sub.2] sin([u.sub.2]), ..., [r.sub.n] cos([u.sub.n]), [r.sub.n] sin([u.sub.n])),

where [u.sub.i] [member of]] R, is called a flat torus in [R.sup.2n]:

By analogy, the subset of [R.sup.2n+1] parameterized by

[??] ([u.sub.1],[u.sub.2], ..., [u.sub.n]) = ([r.sub.1] cos([u.sub.1]), [r.sub.1] sin([u.sub.1]), [r.sub.2] cos([u.sub.2]), [r.sub.2] sin([u.sub.2]), ..., [r.sub.n] cos([u.sub.n]), [r.sub.n] sin([u.sub.n]),a),

where [u.sub.i] [member of] R and a is a real constant, will be called a flat torus in [R.sup.2n+1].

We give the following examples.

Example 1. Any curve in a flat torus of the kind

[alpha](t) = [??]([m.sub.1]t, [m.sub.1]t, ..., [m.sub.n]t)

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.sub.i] [not equal to] [m.sub.j] for all i [not equal to] j. For closed twisted curves see also [13].

Example 2. (Helices in [S.sup.3]) Let [S.sup.3] be the unit 3-sphere imbedded in the Euclidean 4-space [E.sup.4]. A model helix in [S.sup.3] [subset] [E.sup.4] is given by

[gamma](s) = (cos [phi] cos(as), cos [phi] sin(as), sin [phi] cos(bs), sin [phi] sin(bs)),

with

[a.sup.2] [cos.sup.2] [phi] + [b.sup.2] [sin.sup.2] [phi] = 1.

Here s is the arclength parameter. It is easy to see that g lies in the flat torus:

[x.sup.2.sub.1] + [x.sup.2.sub.2] = [cos.sup.2] [phi], [x.sup.2.sub.3] + [x.sup.2.sub.4] = [sin.sup.2] [phi].

Example 3. The Frenet curve [alpha] : I [right arrow] [R.sup.4] given by the parameterization

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a spherical W-curve (with radius 1), (see [11]) where, [r.sub.2.sub.1][m.sup.2.sub.2] + [r.sub.2.sub.2][m.sup.2.sub.1] = [m.sup.2.sub.1][m.sup.2.sub.2]([r.sub.2.sub.1] + [r.sub.2.sub.2]).

3. CURVES OF FINITE TYPE

Let f (s) be a periodic continuous function with period 2[pi]r. Then it is well known that f (s) has a Fourier series expansion given by

f (s) = [a.sub.0]/2 + [a.sub.1] cos s/r + [a.sub.2] cos 2s/r + ... + [b.sub.1] sin s/r + [b.sub.2] sin 2s/r + ...,

where [a.sub.k] and [b.sub.k] are the Fourier coefficients defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [gamma] be a closed curve of length 2[pi]r: If x : g [right arrow] [R.sup.m] is an isometric immersion, then

[x.sup.(j)] = [d.sup.j]x/d[s.sup.j].

Because [DELTA] = - [d.sup.2]/d[s.sup.2] , we have

[[DELTA].sup.j]H = [(-1).sup.j][x.sup.(2j+2)], j = 0,1,2, ... .

If x is of finite type, each coordinate function [x.sub.i] satisfies the following homogeneous ordinary differential equation with constant coefficients:

[x.sup.(2k+2).sub.i] + [c.sub.1][x.sup.(2k).sub.i] + ... + [c.sub.k-1][x.sup.(4).sub.i] + [c.sub.k][x.sup.(2).sub.i] = 0, i = 1,2, ..., m

for some integer k [greater than or equal to] 1 and constant [c.sub.1], ..., [c.sub.k]. Because our solutions [x.sub.i] of the above differential equation are periodic solutions with period 2[pi]r, each [x.sub.i] is a finite linear combination of the following particular solutions:

1, cos ([n.sub.i]s/r), sin ([m.sub.i]s/r), [n.sub.i], [m.sub.i] [member of] Z.

Therefore, each [x.sub.i] is of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some suitable constant [c.sub.i],[a.sub.A](t),[b.sub.A](t) (A = 1, ..., n) and integers [p.sub.A], [q.sub.A]. Thus each [x.sub.i] has a Fourier series expansion of finite sum. Similarly, if each [x.sub.i] has a Fourier series expansion of finite sum, then x is of finite type (see [4,5,7]).

Theorem 1. [3] Let [gamma] be a closed curve of length 2[pi]r in [R.sup.m]. Then isometric immersion x : g [right arrow]] [R.sup.m] is of finite type if and only if the Fourier series expansion of each coordinate function of [gamma],

[gamma](s) = [a.sub.0]+ [[infinity].summation over (t=1)] ([a.sub.t] cos ts/r + [b.sub.t] sin ts/r)

has only finite nonzero terms.

Thus, using the above theorems, we have the following corollary.

Corollary 2. Every closed k-type curve [gamma] in [R.sup.m] can be written in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where T(x) = {[t.sub.1], [t.sub.2], ... [t.sub.k]} is the order of the curve and [a.sub.0],[a.sub.1], ..., [a.asub.k],[b.sub.1], ..., [b.sub.k] are vectors in [R.sup.m] such that for any i in {1,2, ..., k}, [a.sub.i] and [b.sub.i] are not simultaneously zero. Moreover, if q = [t.sub.k] is the upper order of [gamma], then [absolute value of [a.sub.q]] = [absolute value of [b.sub.q]] [not equal to] 0.

Corollary 3. Every null k-type curve [gamma] in [R.sup.m] can be written in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

where [a.sub.0],[b.sub.0],[a.sub.1], ..., [a.sub.k],[b.sub.1], ..., [b.sub.k] are vectors in [R.sup.m] such that [b.sub.0] [not equal to] 0 and for any i in {1,2, ..., k}, [a.sub.i] and [b.sub.i] are not simultaneously zero. Moreover, [absolute value of [a.sub.q]] = [absolute value of [b.sub.q]] [not equal to] 0, where q is the upper order of the curve [gamma].

From (1) and (2) we obtain the following corollary.

Corollary 4. [2]

1) Every k-type curve of [R.sup.m] lies in an affine 2k-subspace [R.sup.2k] of [R.sup.m].

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

4. GENERALIZED HELICES

In the present section we give some well-known definitions of harmonic curvature and Darboux vertex of a curve in [R.sup.m]. We prove that the immersed curve in [R.sup.m] with constant harmonic curvatures [H.sub.i] at point [gamma]([s.sub.0]) has a Darboux vertex at that point.

Definition 2. Let g : I [subset] R [right arrow] [R.sup.m] be a regular curve of rank r with unit speed. For 2 [greater than or equal to] j [greater than or equal to] r-2, the functions [H.sub.j] : I [right arrow] R defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

are called the harmonic curvatures of [gamma], where [[kappa].sub.1],[[kappa].sub.2], ..., [[kappa].sub.r-1] are Frenet curvatures of g which are not necessarily constant and [nabla] is the Levi-Civita connection [12]. For more details see also [1].

Definition 3. The unit speed Frenet curve of rank r is called general helix of order (r-2) if

[r-2.summation over (i=1)][H.sup.2.sub.i] = c, (4)

where c is constant [12].

By the use of (3) and (4) we get the following result.

Proposition 5. Let [gamma] : I [subset] R [right arrow] [R.sup.2n+1] be a regular curve of rank r with unit speed. If [gamma] has constant harmonic curvature, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 4. Let [gamma] be a smoothly immersed curve in [R.sup.2n+1], n > 1, with curvatures [[kappa].sub.1],[[kappa].sub.2], ..., [[kappa].sub.2n-1],[[kappa].sub.2n], where [[kappa].sub.2n] [not equal to] 0. Let us denote

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The Darboux vector in [R.sup.2n+1] is defined by

[??](s) = [a.sub.0]t + [a.sub.1][n.sub.2] + ... + [a.sub.n][n.sub.2]n,

where {t = [gamma]'(s),[n.sub.1],[n.sub.2], ..., [n.sub.2]n} is the Frenet frame of [gamma] [15].

Lemma 6. [15] The derivative of [??](s) is

[??]'(s) = [a'.sub.0]t + [a'.sub.1][n.sub.2]+ ... +[a'.sub.n][n.sub.2]n.

Definition 5. (Darboux vertex): The point [gamma]([s.sub.0]) is called Darboux vertex of [gamma] if the first derivative of the Darboux vector [??] (s) is vanishing at that point.

Theorem 7. [15] Let [gamma] be a smoothly immersed curve in [R.sup.2n+1](n[greater than or equal to]1), with [[kappa].sub.1],[[kappa].sub.2], ..., [[kappa].sub.2n] for its curvatures. The curve has a Darboux vertex at point [gamma]([s.sub.0]) if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

Proof. Let [gamma] be a smoothly immersed curve in [R.sup.2n+1]: If g has a Darboux vertex at [gamma]([s.sub.0]), then by Lemma 6 we get [a'.sub.0] = 0, [a'.sub.1] = 0, ..., [a'.sub.n] = 0: By Definition 4 we get the result.

By Proposition 5 and Theorem 7 we get the following results.

Corollary 8. Let [gamma] : I [subset] R [right arrow][R.sup.2n+1] be a regular curve of rank 2n with unit speed. If the harmonic curvatures [H.sub.i] are constant at the point [gamma]([s.sub.0]), then g has a Darboux vertex at that point.

Proof. If the harmonic curvatures [H.sub.i] are constant at the point [gamma]([s.sub.0]), then by Proposition 5 all the ratios [[kappa].sub.1]/[[kappa].sub.2], [[kappa].sub.3]/[[kappa.sub.4], ..., [[kappa].sub.2n-1]/[[kappa].sub.2n] are constant. So, taking the derivatives of the ratios [[kappa].sub.1]/[[kappa].sub.2], [[kappa].sub.3]/[[kappa].sub.4], ..., [[kappa].sub.2n-1]/[[kappa].sub.2n] with respect to s, we obtain (5). Using Theorem 7, we complete the proof.

Corollary 9. If [gamma] : I [subset] R [right arrow] [R.sup.2n+1] has a Darboux vertex at the point [gamma]([s.sub.0]), then g is a general helix of order (2n-1).

5. CURVES WITH CONSTANT CURVATURE RATIOS

A curve [gamma] = [gamma](s) : I [right arrow] [R.sup.m] is said to have constant curvature ratios (ccr-curve) if all the quotients [k.sub.i+1]/[k.sub.i] are constant [11].

As is well known, generalized helices in [R.sup.3] are characterized by the fact that the quotient [[tau].sub.k] is constant (Lancret's theorem). It is in this sense that ccr-curves are generalization to [R.sup.m] of generalized helices in [R.sup.3].

In [9] a generalized helix in the m-dimensional space (m odd) is defined as a curve satisfying that the ratios [[kappa].sub.2]/[[kappa].sub.1], [[kappa].sub.4]/[[kappa].sub.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 [beta] be a ccr-curve with non-constant curvature. Then Frenet's formulae of [beta] are reduced to a linear system of first order differential equations with constant coefficients

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

for some constants [c.sub.2], ..., [c.sub.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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [u.sub.i] are the eigenvectors, [[lambda].sub.i] are the eigenvalues of the constant coefficient matrix of the system, and [d.sub.i] are arbitrary constants.

We prove the following main result.

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

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.sub.n-1] = 0.

From now on, we shall consider that all the curvatures, and all the constants [c.sub.i] are not zero. Therefore, the eigenvalues are all of multiplicity 1.

Let [[lambda].sub.l] = [[alpha].sub.l] [+ or -] i[[mu].sub.l], l = 1, ..., [n/2], with [[alpha].sub.l], [[mu].sub.l] [member of] 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and similarly for n = 2k+1, the general solution of the system for the first vector is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [a.sub.0], .[u.sub.1], ..., [u.sub.k] are vectors in [R.sup.m] and [d.sub.l], [f.sub.l] are arbitrary constants.

Condition [parallel][[??].sub.1](u)[parallel] = 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

[[??].sub.1](u) = [k.summation over (l=1)][d.sub.l][u.sub.l] cos([[mu].sub.l]u)+ [f.sub.l][u.sub.l] sin([[mu].sub.l]u).

Similarly for n = 2k+1, the general solution of the system for the first vector is

[[??].sub.1](u) = [a.sub.0] + [k.summation over (l=1)] [d.sub.l][u.sub.l] cos([[mu].sub.l]u) + [f.sub.l][u.sub.l] sin([[mu].sub.l]u),

where [a.sub.0],[u.sub.1], ..., [u.sub.k] are vectors in [R.sup.m] and [d.sub.l], [f.sub.l] are arbitrary constants.

Since [beta] (u) = [[??].sub.1](u); for n = 2k;

[beta] (s) = [a.sub.0] + [k.summation over (l=1)] [[??].sub.l] cos([[mu].sub.l]s) + [[??].sub.l] sin([[mu].sub.l]s).

Similarly for n = 2k+1,

[beta] (s) = [b.sub.0] + [a.sub.0]s + [k.summation over (l=1)] [[??].sub.l] cos([[mu].sub.l]s) + [[??].sub.l] sin([[mu].sub.l]s),

where [[??].sub.l] = [d.sub.l]/[[mu].sub.l][u.sub.l] and [[??].sub.l] = - [f.sub.l]/[[mu].sub.l] [u.sub.l] are vectors which are not necessarily constant. So, using Corollary 2 and Corollary 3, we complete the proof of the theorem. []

Example 4. The Frenet curve [alpha] : I [right arrow] [R.sup.4] given by the parameterization

[alpha](s) = (0, - [square root of 3]/2, 0, 1/2) + [[integral].sup.s.sub.0] [[??].sub.1](arcsin(2u))du, s [member of]] - 1/2, 1/2[

is a spherical ccr-curve with the centre at the origin of coordinates, with radius 1 and non-constant curvatures (see [11]), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Received 14 August 2008, in revised form 11 September 2008

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[16.] Weisstein, E. W. Ordinary Differential Equation--System with Constant Coefficients. From Math-World-A Wolfram Web Resource. http://mathworld.wolfram.com/Ordinary Differential Equation System with Constant Coefficients.html (accessed 15 Sept. 2008).

Gunay Ozturk (a), Kadri Arslan (b) *, and H. Hilmi Hacisalihoglu (c)

(a) Department of Mathematics, Kocaeli University, 41380 Kocaeli, Turkey; [e-mail]

(b) Department of Mathematics, Uludag University, 16059 Bursa, Turkey

(c) Department of Mathematics, Ankara University, 06100 Ankara, Turkey; [e-mail]

* Corresponding author, [e-mail]