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Council for Innovative Research Peer Review Research Publishing System Journal: Journal of Advances in Mathematics View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by KHALSA PUBLICATIONS ISSN 2347-1921 Some Geometrical Calculations for the Spherical Indicatrices of Involutes of a Timelike Curve in Minkowski 3-Space Mustafa Bilici, Mustafa Çalışkan Ondokuz Mayıs University, Educational Faculty, Department of Mathematics 55200 Kurupelit, Samsun, Turkey [email protected] Gazi University, Faculty of Sciences, Department of Mathematics, 06500 Teknik Okullar,Ankara, Turkey [email protected] ABSTRACT In this paper, we introduce the spherical indicatrices of involutes of a given timelike curve. Then we give some important 3 2 2 relationships between arc lengths and geodesic curvatures of the base curve and its involutes on E1 , S1 , H0 . Additionally, some important results concerning these curves are given. Keywords: Minkowski space; involute evolute curve couple; spherical indicatrices; geodesic curvature. Mathematics Subject Classification 2010: 51B20, 53B30, 53C50 Council for Innovative Research Peer Review Research Publishing System Journal: Journal of Advances in Mathematics Vol 5, No. 2 [email protected] www.cirworld.com, member.cirworld.com 668 | P a g e D e c 2 3 , 2 0 1 3 ISSN 2347-1921 INTRODUCTION The notion of the involute of a curve was introduced by Christiaan Huygens in his work titled Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae (1673). In the theory of curves in Euclidean space, one of the important and interesting problems is the characterizations of a regular curve. In particular, the involute of a given curve is a well known concept in the classical differential geometry (see [5]). At the beginning of the twentieth century, A. Einstein’s theory opened a door of use of new geometries. One of them is simultaneously the geometry of special relativity and the geometry induced on each fixed tangent space of an arbitrary Lorentzian manifold, was introduced and some of classical differential geometry topics have been treated by the researchers. According to reference ([1], [2], [3]), it has been investigated the involutes of the timelike curves in Minkowski 3-space. Also, Bükcü and Karacan studied on the involute and evolute curves of the spacelike curve with a spacelike binormal in Minkowski 3-space, [4]. In this study, we carry tangents of the spacelike involute of a timelike curve to the center of the pseudohyperbolic space and we obtain a spacelike curve with equation on . This curve is called the first spherical indicatrix or tangent indicatrix of . Similarly one consider the principal normal indicatrix and the3 binormal2 2 E1 S1 H0 indicatrix on the pseudosphere . Then we give some important relationships between arc lengths and geodesic curvatures of the base curve and its involutes on , , . Additionally, some important results concerning these curves are given. PRELIMNARIES The Minkowski 3-space is the vector space E3 provided with the Lorentzian inner product g given by 222 g(,) X X x1 x 2 x 3 , 3 where X x1 ,x 2 ,x 3 E . A vector X is said to be timelike if g X ,X 0 , spacelike if g X ,X 0 and lightlike (or null) if g X ,X 0 . Similarly, an arbitrary curve s in where s is a pseudo-arclenght parameter, can locally be timelike spacelike or null (lightlike), if all of its velocity vectors s are respectively timelike, spacelike or null. The norm of a vector is defined by X g X ,X 3 and two vectors X x1 ,x 2 ,x 3 , Y y1 ,y 2 ,y 3 E 1 are orthogonal if and only if g X ,Y 0 . Now let and Y be two vectors in , then the Lorentzian cross product is given by e1 e 2 e 3 XY x1 x 2 x 3 xy 32 xy,xy 2313 xy,xy 3112 xy 21 , [6]. y1 y 2 y 3 Denote by T s ,N s ,B s the moving Frenet frame along the curve in the space . Then T , N and B are the tangent, the principal normal and the binormal vector of the curve , respectively. For these vectors, we can write TNB,NBT,BTN . Depending on the causal character of the curve , the following Frenet-Serret formulas are given in [7]. T N, N T B, B N . gT,T 1,gN,N gB,B 1,gT,N gT,B gN,B 0 If the curve is non-unit speed curve in space , then det , , , . 3 2 The Darboux vector for the timelike curve is defined by [8] 669 | P a g e D e c 2 3 , 2 0 1 3 ISSN 2347-1921 TB . There are two cases corresponding to the causal characteristic of Darboux vector . Case I. If κτ , then is a spacelike vector. In this situation, we can write cosh 2 22 ,,g sinh and the unit vector C of direction is 1 CTB sinh cosh , where is the lorentzian timelike angle between B and unit timelike vector C that lorentz orthogonal to the normalisation of the Darboux vector (see Figure 1). Fig 1: Lorentzian timelike angle Case II. If κτ , then is a timelike vector. In this situation, we have sinh 2 22 ,, g cosh and the unit vector of direction is 1 CTB cosh sinh . τ τ Remark 1. We can easily see from equations of the section Case I and Case II that: = tanhφ or = cothφ , if κ κ constant then is a general helix. For the arc length of the spherical indicatrix of C we get 670 | P a g e D e c 2 3 , 2 0 1 3 ISSN 2347-1921 s s ds . C 0 After some calculations, we have for the arc lenghts of the spherical indicatrices T,N,B measured from the points corresponding to s 0 s s s s ds,s ds,s ds TNB 0 0 0 for their geodesic curvatures with respest to 11 kkTT, for timelike cosh sinh 1122 kk22 , for timelike 3 NNE1 11 kkBB, for timelike sinh cosh 22 k 1 k 1 , for timelike CC 2 2 and for their geodesic curvatures with respest to S1 or H0 TTT tT t tanh coth , for timelike t N tN N W , [6]. t coth tanh , for timelike B tB B B W t C tC C 2 2 Note that and are Levi-Civita connections on S1 and H0 , respectively. Then Gauss equations are given by the followings. XXXY YgSXY ,,,,, YX YgSXY g , 2 2 where is a unit normal vector field and S is the shape operator of S1 (or H0 ). 3 The unit pseudosphere and pseudohyperbolic space of radius 1and center 0 in E1 are given by 23 S1 X x 1, x 2 , x 3 E 1 : g X , X 1 and 23 H0 X x 1, x 2 , x 3 E 1 : g X , X 1 respectively, [4]. *3 Definition 1. Let s , s E1 be two curves. Let Frenet frames of and be T,N,B and T,N,B*** , respectively. is called the involute of ( is called the evolute of ) if g T ,T* 0 . 671 | P a g e D e c 2 3 , 2 0 1 3 ISSN 2347-1921 Remark 2. Let be a timelike curve. In this situation its involute curve must be a spacelike curve. , being the involute-evolute curve couple, the following lemma was presented by Bilici and Çalışkan ( see [1], [3]). Lemma 1. Let , be the timelike-spacelike involute-evolute curve couple. The relations between the Frenet vectors of the curve couple as follow. (1) If is a spacelike vector( ), then T* 0 1 0 T * Ncosh 0 sinh N . * Bsinh 0 cosh B (2) If is a timelike vector( ), then T* 0 1 0 T * Nsinh 0 cosh N . * Bcosh 0 sinh B Example 1. Let the curve s 2sinh s , 2 cosh s , s be a unit speed timelike curve such that 2 s 2 s 1 Ts cosh , sinh , . 3 3 3 3 3 If is a timelike curve then its involute curve is a spacelike curve. In this situation, involute curve of the curve can be given the equation 2 s 2 s 2 s 2 s c s sinh c s cosh , cosh c s sinh , , 3 3 3 3 3 3 3 3 3 where c is an arbitrary constant. If we compute the tangent vector field of the curve we have * T s sinh ,cosh , 0 33 and g T s ,T* s 0 . 3. SOME GEOMETRICAL CALCULATIONS FOR THE SPHERICAL INDICATRICES OF INVOLUTES OF A TIMELIKE CURVE In this section, we compute the arc-lengths of the spherical indicatrix curves T,N,B*** and then we calculate the 3 2 2 geodesic curvatures of these curves in E1 and S1 (or H0 ). Firstly, for the arc-length s of tagent indicatrix T * of the involute curve , we can write T* * s * S dT dN s ds* ds , T* * 00ds ds S 22 s* ds T 0 672 | P a g e D e c 2 3 , 2 0 1 3 ISSN 2347-1921 or from the definition of Darboux vector S s* ds . T 0 If the arc length for the principal normal indicatrix N* is s it is N* SSdN * dcosh T sinh B s* ds ds , N 00ds ds S 2 2 s* ds . N 0 For timelike , from the Lemma1. (2) we have S 2 2 s* ds . N 0 the arc length for the binormal indicatrix B* is s it is If B* SSdB* dsinh T cosh B s* ds ds , B 00ds ds S s* ds . B 0 If is a timelike vector, then we have the same result.
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