Selçuk J. Appl. Math. Selçuk Journal of Vol. 14. No. 1. pp. 83-87, 2013 Applied Mathematics
e-Curvature Functions inR3 1
Esen ·Iyigün
Uluda¼g University, Art And Science Faculty, Department of Mathematics, 16059, Bursa, Turkiye.
e-mail:esen@ uludag.edu.tr
Received Date: August 27, 2012 Accepted Date: September 03, 2012
Abstract. In this study; we de…ne the ith e-curvature functions mi for 1 i 3 in R31:So, in R31 we give a relation between harmonic curvatures and ith e-curvature functions. Moreover, we …nd a relation between constant e-curvature ratios and ithe-curvature functions. Finally, we apply on a unit speed time-like curve to the some given results.
Key words: ith e-curvature function m
i; ccr-curve; 3-dimensional Lorentz Minkowski space R3
1:
AMS Classi…cation: 53C40, 53C42. 1. Introduction
Let X = (x1; x2; x3) and Y = (y1; y2; y3) be two non-zero vectors in 3-dimensional Lorentz Minkowski space R31:We denoted R31 shortly by L3:For X; Y 2 L3
hX; Y i = x1y1+ x2y2+ x3y3 is called Lorentzian inner product. The couple R3
1; h; i is called Lorentzian space and brie‡y denoted by L3.Then the vector X of L3is called
i)time-like if hX; Xi < 0,
ii)space-like if hX; Xi > 0 or X = 0,
iii)null (or light-like) vector if hX; Xi = 0, X 6= 0.
Similarly, an arbitrary curve = (s) in L3 can be locally be space-like, time-like or null, if all of its velocity vectors 0(s) are respectively space-like, time-like or null.Also, recall the norm of a vector X is given by kXk =pjhX; Xij.Therefore, X is a unit vector if hX; Xi = 1:Next, vectors X; Y in L3 are said to be or-thogonal if hX; Y i = 0:The velocity of the curve is given by 0 .Thus, a
space-like or a time-like is said to be parametrized by arclength function s;if D 0
; 0E= 1 [1]. 2. Basic De…nitions
De…nition 1. Let : I ! L3be a curve in L3and k
1; k2the Frenet curvatures of . Then for a unit tangent vector V1 =
0
(s) over M the ith e-curvature function mi , 1 i 3 is de…ned by mi = 8 > > > < > > > : 0 ; i = 1 "1"2 k1 ; i = 2 d dt(mi 1) + "i 2mi 2ki 2 "i ki 1 ; 2 < i 3 9 > > > = > > > ; where "i= hVi; Vii = 1.
De…nition 2. Let : I ! L3be a unit speed non-null curve in L3: The curve is called Frenet curve of osculating order d , (d 3) if its 3rdorder derivatives
0
(s); 00(s); 000(s); are linearly independent and 0(s); 00(s); 000(s); {v(s) are no longer linearly independent for all s 2 I: For each Frenet curve of order 3 one can associate an orthonormal 3 frame fV1; V2; V3g along (such that
0
(s) = V1) called the Frenet frame and the Frenet formulas is de…ned in the usual way; V10 = rv1 0 = "2k1V2; V20 = rv1V2= "1k1V1+ "3k2V3; V30 = rv1V3= "2k2V2: 3. Harmonic Curvatures
De…nition 3. Let be a non-null curve of osculating order 3.The harmonic functions Hj: I ! R ; 0 j 1; de…ned by 8 < : H0= 0; H1= k1 k2 = "1"2"3 m3 m2(m2)0
are called the harmonic curvatures of , where the ith e-curvature function mi , 1 i 3:
De…nition 4. Let be a time-like curve in L3 with 0
(s) = V1. X L3 being a constant unit vector …eld if
then is called a general helix (inclined curves) in L3, ' is called slope angle and the space Sp fXg is called slope axis [7].
De…nition 5. Let be a non-null of osculating order 3. Then is called a general helix of rank 1 if
H12= c; holds, where c 6= 0 is a real constant. We have the following results.
Corollary 1. i) If H1= 0 then is a straight line.
ii) If H1is constant then is a general helix of rank 1. Proof. By the use of above de…nition we obtain.
Proposition 1. V10 = rv1 0 = "1 m2 V2; V20 = rv1V2= "2 m2 V1+ (m2)0 m3 V3; V30 = rv1V3= "2"3 (m2) 0 m3 V2:
Proof. By using de…nition of the ithe-curvature function m
i, 1 i 3 ,we get the result.
4. Constant Curvature Ratios
De…nition 6. A curve : I ! L3 is said to have constant curvature ratios (that is to say, it is a ccr-curve) if all the quotients "i
ki+1 ki
are constant. Here; ki; ki+1are Frenet curvatures of and "i = hVi; Vii = 1:
Corollary 2. For i=1 ccr-curve is "2"3
m2(m2)
0
m3 :
Proof. The proof can be easily seen by using the de…nitions of the ith e-curvature function mi and ccr-curve.
Corollary 3. Let : I ! L3 is a ccr-curve. If " 2"3 m2(m2) 0 m3 ! = constant; then "2"3 m2(m2) 0 m3 !0 = 0:
Proof. The proof is obvious.
Theorem 1. is a ccr-curve in L3, "
1H12= constant:
Proof. By using the de…nitions of a general helix of rank 1 and ccr-curve, this completes the proof of the theorem.
Now, we will calculate constant curvature ratios and the ithe-curvature function mi, 1 i 3, of a unit speed time-like curve in L3.
5. An Example
Example 1. Let us consider the following curve in the space L3 (s) = p2s; sin s; cos s :
V1(s) =
0
(s) = p2; cos s; sin s
whereD 0(s); 0(s)E= 1; which shows (s) is a unit speed time-like curve.Thus
0
(s) = 1:We express the following di¤erentiations:
00 (s) = (0; sin s; cos s) ) 000 (s) = (0; cos s; sin s) and V2(s) = "2 00 (s) k 00(s)k = 00 (s): So, we have the …rst curvature as
k1(s) = D
V10(s); V2(s) E
= 1 = constant. Moreover we can write third Frenet vectors of the curve,
V3(s) = V1(s) V2(s) = 1; p
2 cos s; p2 sin s : Finally, we have second curvature of (s) as
k2(s) = D
V20(s); V3(s) E
= p2 = constant. Now, we will calculate ithe-curvature function m
i; 1 i 3 , and ccr-curve of (s) in L3: m2= "1"2 k1 = 1; m3= "3(m2) 0 k2 = 0; "1 k2 k1 = p2 = constant:
Thus, (s) is a ccr-curve in L3: References
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