Characterizations of some special helices in [E.sup.4].

In this work, notion of slant helix concept in  is extented to the space [E.sup.4]. With an analogous way, we define 3--type slant helices whose trinormal lines make a constant angle with a fixed direction in [E.sup.4]. Moreover, some characetrizations of such curves and other forms (in the case of principal lines or binormal lines make a constant angle with a fixed direction) of slant helices are presented.

Keywords Euclidean space, frenet frame, slant helices.

[section] 1. Introduction

It is safe to report that many important results in the theory of the curves in [E.sup.3] were initiated by G. Monge; and G. Darboux pionnered the moving frame idea. Thereafter, F. Frenet defined his moving frame and his special equations which play important roles in mechanics and kinematics as well as in differential geometry (for more details see ).

In the case of a differentiable curve, at each point a tetrad of mutually orthogonal unit vectors (called tangent, normal, binormal and trinormal)was defined and constructed, and the rates of change of these vectors along the curve define the curvatures of the curve in the space [E.sup.4] . And this tetrad and curvatures are called all together 'Frenet Apparatus' of the curves. And helices (inclined curves) are well known concepts in the classical differential geometry . Recall that an arbitrary curve is called a W--curve, if it has constant Frenet curvatures .

The notion of slant helix is due to Izumiya and Takeuchi . They defined that a curve [phi] = [phi](s) with non-vanishing first curvature is called a slant helix in [E.sup.3] if the principal lines of [phi] make a constant angle with a fixed directon. In this paper, we investigated slant helices and other cases (if the binormal lines or trinormal lines make a constant angle with a fixed direction) and gave some characterizations of mentioned curves in the space [E.sup.4].

[section] 2. Preliminaries

To meet the requirements in the next sections, here, the basic elements of the theory of curves in the space [E.sup.4] are briefly presented (A more complete elementary treatment can be found in ).

Let [alpha] : I [subset] R [right arrow] [E.sup.4] be an arbitrary curve in the Euclidean space [E.sup.4]. Recall that the curve [alpha] is said to be of unit speed (or parametrized by arclength function s) if <[alpha]'(s), [alpha]' (8)> = 1, where <.,.> is the standart scalar (inner) product of [E.sup.4] given by

(X, Y) = [x.sub.1][y.sub.1] + [x.sub.2][y.sub.2] + [x.sub.3][y.sub.3] + [x.sub.4][y.sub.4],

for each X = ([x.sub.1], [x.sub.2], [x.sub.3], [x.sub.4]), Y = ([y.sub.1], [y.sub.2], [y.sub.3], [y.sub.4]) [member of] [E.sub.4]. In particular, the norm of a vector X [member of] [E.sup.4] is given by

||X|| = [square root of (<X, X>)].

Let {T(s), N(s), B(s), E(s)} be the moving frame along the unit speed curve [alpha], where T, N, B and E denote, respectively the tangent, the principal normal, the binormal and the trinormal vector fields. Then the Frenet formulas are given by 

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

The real valued functions [kappa], [tau] and [sigma] are called, respectively, the first, the second and the third curvature of [alpha]. If [sigma] [not equal to] 0 for each s [member of] I [subset] R, the curve lies fully in [E.sup.4]. Recall that the unit sphere [S.sup.3] in [E.sup4], centered at the origin, is the hyper surface defined by

[S.sup.3]={X[member of][E.sup.4]:<X,X>=1}.

In this work, we shall assume that [kappa] [not equal to] 0, [tau] [not equal to] 0 and [sigma] [not equal to] 0 for each s [member of] I [subset of] R.

[section] 3. Characterizations of Some Special Helices in [E.sup.4]

Theorem 1. There isn't any curve in [E.sup.4] such that;

i) The principal normal lines of it make a constant angle with a fixed direction.

ii) The binormal lines of it make a constant angle with a fixed direction.

Proof.

i) Let us suppose there is a curve which is defined with statement i. Thus, we can write that

N. U = cos w, (2)

where U is a constant vector (fixed direction) and w is a constant angle. Differentitating both sides of (2) and considering Frenet equations, we get

(-[kappa]T +[tau]B).U = 0. (3)

(3) implies that T [perpendicular] U and B [perpendicular] U. Therefore we can compose U as

U =- [u.sub.1]N + [u.sub.2]E. (4)

Differentiating (4), we easily have

[u.sub.1] = cosw = [u.sub.2] = 0, (5)

ii) Similar to above proof, let us assume that there is a curve which hold statement ii. Now, we write

B.U = cos[xi] (6)

where U is a constant vector (fixed direction) and [xi] is a constant angle. Following same procedure in proof of i, we have components of U zero, respectively. This result yields a contradiction.

Definition. A curve [psi] = [psi](s) is called a 3-type slant helix if the trinormal lines of [psi] make a constant angle with a fixed direction in [E.sup.4].

Theorem 2. Let [psi] = [psi](s) be a 3-type slant helix with non-vanishing curvatures in [E.sup.4].

i) There is a relation among curvatures of [psi] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

ii) Fixed direction of this helix can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

where [delta] [not equal to] [kappa] [pi]/2 is a constant angle and A and B are real numbers.

Proof.

1) From definition, we write

E. U = cos[delta], (9)

where U is a fixed direction and [delta] [not equal to] [kappa] [pi]/2 is a constant angle. Differentiating both sides of (9), we have

-[sigma]B.U = 0. (10)

And therefore, we compose constant vector U as

U = [u.sub.1]T + [u.sub.2]N + [u.sub.3]E. (11)

Differentiating both sides of (11) respect to s and considering Frenet equations, we have a system of differential equation as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

Using [(12).sub.4], we have

[u.sub.3] = cos [delta] = constant. (13)

Here, [u.sub.3] can not be zero. If it is, then U lies fully in TN hyperplane and it follows that [u.sub.1] = [u.sub.2] = 0. Thereby [u.sub.3] [not equal to] 0. Substituting (13) to [(12).sub.3], we have

[u.sub.2] = cos [delta]*[sigma]/[tau]. (14)

Using equations [(12).sub.1] and [(12).sub.2], we have second order differential equation as follow:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)

Using exchange variable t = [??] [kappa]ds in (15), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (16)

Solution of (16) gives us the first component

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)

where A, B are real numbers. And we easily have the second component

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (18)

(18) completes proof of 1).

ii) Using obtained equations we write fixed direction as follow:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (18)

Corollary. The third curvature of 3-type slant helix in [E.sub.4] can not be zero. Therefore, [psi] never lies in TNB subspace.

Theorem 3. 3-type slant helix with non-vanishing curvatures can not be a W--curve in [E.sup.4].

Proof. Let us suppose 3-type slant helix with non-vanishing curvatures is a W--curve in [E.sup.4]. In this case, if we consider solution of (12), we have

[u.sub.1] = [u.sub.2] = [u.sub.3] = 0, (20)

References

 C.Boyer, A History of Mathematics, New York, Wiley, 1968.

 H.Gluck, Higher Curvatures of Curves in Euclidean Space, Amer. Math.Monthly, 73 (1996), 699-704.

 H.H.Hacisalihoglu, Differential Geometry, Ankara University Faculty of Science, 2000.

 K. Ilarslan and O. Boyacioglu, It Position Vectors of a Space-like W-curve in Minkowski Space [[E.sup.3]sub.1], Bull. Korean Math. Soc., 44(2007), 429-43.

 R. S. Milman and G. D. Parker, Elements of Differential Geometry, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1977.

 S.Izumiya and N.Takeuchi, New Special Curves and Developable Surfaces, Turk J. Math., 28(2004), 153-163.

Melih Turgut and Suha Yilmaz

Dokuz Eylul University, Buca Educational Faculty, Department of Mathematics,

35160 Buca-Izmir, TURKEY

E-mail: melih.turgutUgmail.com

E-mail: suha.yilmazUyahoo.com
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