On harmonic curvatures of null generalized helices in L4

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Selçuk J. Appl. Math. Selçuk Journal of Vol. 13. No. 2. pp. 63-67, 2012 Applied Mathematics

On Harmonic Curvatures of Null Generalized Helices in L4 Esen ·Iyigün

Uluda¼g University, Faculty of Arts And Science, Department of Mathematics, 16059, Bursa,Turkiye

e-mail:esen@ uludag.edu.tr

Received Date: October 19, 2011 Accepted Date: December 11, 2012

Abstract. In this study; we give a relation between harmonic curvatures and the Frenet equations of a null curve in a 4 dimensional Lorentz space. Also, we obtain some theorems and we give an example of a null helix.

Key words: Null curve; Harmonic curvature; Null Frenet curve of osculating order 4; Null helix.

2000 Mathematics Subject Classi…cation: 53C40, 53C42. 1. Introduction

Let x = (x1; x2; x3; x4) and y = (y1; y2; y3; y4) be non-zero vectors in Minkowski 4 space R4 1: We denoted R41 by L4: For x; y 2 L4 hx; yi = x1y1+ 4 X i=2 xiyi

is called Lorentzian inner product. The couple R41; h; i is called Lorentzian space and denoted by L4 . Then the vector v of L4 is called i) time-like if hv; vi < 0, ii) space-like if hv; vi > 0 or v = 0, iii) null (or light-like) vector if hv; vi = 0, v 6= 0. An arbitrary curve = (t) in L4_{can be locally be space-like,} time-like or null ( light-like), if all of its velocity vectors 0(t) are respectively space-like, time-like or null, [10].

2. Basic De…nitions

De…nition 1. Let : I _{! L}4 _{be a null curve in L}4: The curve is called a Frenet curve of osculating order 4, if its 4thorder derivatives 0(t); 00(t); 000(t);

{v_{(t) are linearly independent and} 0

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longer linearly independent for all t 2 I: For each null Frenet curve of osculating order 4; one can associate an orthonormal 4 frame fT; N; W1; W2g along (such that 0_{(t) = T ) called the Frenet frame and functions fk}1; k2; k3; k4g called the Frenet curvatures. Thus from [1], the Frenet equations of a null curve in a 4-dimensional Lorentz manifold are written down as follows:

8 > > < > > : rTT = hT + k1W1 rTN = hN + k2W1+ k3W2 rTW1= k2T k1N + k4W2 rTW2= k3T k4W1; where r is the Levi-Civita connection of L4_{, h and fk}

1; k2; k3; k4g are di¤erential functions, T and N are null vectors , W1and W2are space-like vectors. In these equations, by changing a suitable parameter t, we may take h = 0 and other equations stay unchanged. This parameter is called distinguished parameter of the curve [1]. That is,

(1) 8 > > < > > : rTT = k1W1; rTN = k2W1+ k3W2; rTW1= k2T k1N + k4W2; rTW2= k3T k4W1:

From [1] again, since T and N are null vectors, Wi ; 1 i 2, are space-like vectors then we have

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hT; T i = hN; Ni = hT; W1i = hT; W2i = hN; W1i = hN; W2i = hW1; W2i = 0; hT; Ni = 1 and hW1; W1i = hW2; W2i = 1:

De…nition 2. If a null curve : I _{! L}4 is a null Frenet curve of osculating order 4 and Frenet curvatures ki; 1 i 4 are non-zero constants, then is called a null W curve of rank 4:

3. Null Generalized Helix in L4

De…nition 3. [2] Assume that L4 is a null generalized helix given by curvature functions k1; k2; k3: Then the harmonic curvatures of in L4 are written down as follows:

(3) Hi= 8 > > < > > : k2 k1 ; i = 1 H10 k3 ; i = 2:

De…nition 4. Let _{be a time-like curve in L}4 _with 0

(s) = V1. Let X L4 be a constant unit vector …eld. If

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then _{is called a general helix (inclined curve) in L}4, ' is called the slope angle and the space Sp fXg is called slope axis [9].

De…nition 5. [2] A null curve : I _{! L}4 _{is said to be a generalized} helix if there exist a non-zero unit constant vector X such thatD 0(t); XE ₆₌ 0; is cons tan t: Then Sp fXg is called slope axis and for the Frenet frame fT; N; W1; W2g we have

hN; Xi = H1hT; Xi ; hW1; Xi = 0; hW2; Xi = H2hT; Xi :

Theorem 1. Let _{be a null Frenet curve of osculating order 4 in L}4: Then

(4) k4=

k2+ k1H1 H2

;

where k1; k2; k4 are the Frenet curvatures and H1; H2 are harmonic curvatures of :

Proof. By the use of a previous de…nition we obtain hrTW1; Xi = 0 ) h k2T k1N + k4W2; Xi = 0

) k2hT; Xi k1hN; Xi + k4hW2; Xi = 0 ) k2hT; Xi k1H1hT; Xi + k4H2hT; Xi = 0 ) hT; Xi ( k2 k1H1+ k4H2) = 0;

where, since we know that hT; Xi 6= 0; is cons tan t;we can write k2 k1H1+ k4H2= 0: Thus, we obtain k4= k2+ k1H1 H2 :

Theorem 2. Let _{be a null Frenet curve of osculating order 4 in L}4_{: Then} 8 > > > > > > > > < > > > > > > > > : rTT = k1W1 rTN = k1H1W1+ H₁0 H2 W2 rTW1= k1H1T k1N + k2+ k1H1 H2 W2 rTW2= H₁0 H2 T k2+ k1H1 H2 W1;

where k1; k2are Frenet curvatures and H1; H2are harmonic curvatures of ; r is the Levi-Civita connection of L4:

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Proof. By using; (1),(3) and (4), we obtain the proof of the theorem. Theorem 3. Let : I _{! L}4_{be a null curve in L}4_{: Then}

8 > > > > > > > < > > > > > > > : hrTW1; W1i = hrTT; W2i = hrTW2; W2i = 0; hrTW2; W1i = hrTW1; W2i ; hrTN; W2i = H₁0 H2 ; hrTN; W1i = H1k1; hrTT; W1i = k2 H1 :

Here; T and N are null vectors , W1 and W2 are space-like vectors, H1and H2 are harmonic curvatures of _{; r is the Levi-Civita connection of L}4 _{and k}

1; k2 are Frenet curvatures of :

Proof. By using; (1), (2) and (3), we obtain the proof of the theorem. 4. Example

Example 1. Let : I _{! L}4be the null curve de…ned by (t) = (sinh t; cosh t; t; 0)

and X = (0; 0; 1; 0) be a unit constant vector …eld in L4_{: The tangent vector of} is

T = 0(t) = (cosh t; sinh t; 1; 0)

and hT; T i = 0, so is a null curve in L4_{: Also, hT; Xi = 1 = constant: Therefore} the curve is a null helix.

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