A NEW CHARACTERIZATION FOR INCLINED CURVES BY THE HELP OF SPHERICAL REPRESENTATIONS

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1 International Electronic Journal of Geometry Volume 2 No. 2 pp (2009 c IEJG A NEW CHARACTERIZATION FOR INCLINED CURVES BY THE HELP OF SPHERICAL REPRESENTATIONS H. HILMI HACISALIHOĞLU (Communicated by Yusuf YAYLI Abstract. In this wk, arc lengths of spherical representations of tangent vect field T, principal nmal vect field N, binmal vect field B and the vect fiel = W, where W = T + κ B is the Darboux vect field of a W space curve α in E 3 are calculated. Let us denote the spherical representation of T ( T ( N ( B ( C, N, B and C by,, and, respectively. ( C The arc element c of the spherical representation expressed in terms of the harmonic curvature H = κ. Thus the following characterization is given. The curve α E 3 is an inclined curve ( if and only if the arc length s c of C the Darboux spherical representation of α is constant. 1. Introduction In recent years, many imptant and intensive studies are seen about inclined curves. Papers in [1], [2],..., [21] show that how imptant field of interest inclined curves have. Let κ and be the curvatures of a curve in E 3 İn the generalization to E n, n 3, they consider the following cases: (a κ = e te and = e te, (b κ e te and e te, but H = κ = ete. The case (a f the generalization to E n is not seen to be interesting. However, by generalizing the harmonic curvature H = κ to En, the wks in (b are me interesting [13], [18], [19]. F this reason, we have given a new characterization f the inclined curves which satisfy the case (b. This comes into light by means of spherical representations of α. 2. Characterizations f Ordinary Helices and Inclined Curves 2.1. The arc length of tangentian representation of the curve α E 3. Let T = T (s be the tangent vect field of the curve 2000 Mathematics Subject Classification. 53A04. Key w and phrases. Inclined curve, harmonic curvature, dinary helix. 71

2 72 H. HILMI HACISALIHOĞLU s α(s. The spherical curve α T = T on S 2 is called I.st spherical representation of the tangents of α. Let s be the arc length parameter of α. If we denote the arc length of the curve α T by s T, then we may write α T (s T = T (s. Letting dα T T = T T we have T T = κ N T.Hence we obtain T = κ. Thus we give the following result. If κ is the first curvature of the curve α : I E 3, then the arc length s T of the tangentian representation α T of α is s T = κ + c. If the harmonic curvature of α is H = κ, we get T = H + c where c is an integral constant. Thus we have the following theem. Theem 2.1. α E 3 is an dinary helix if and only if s T = Hs + c The Arc Length of the Principal Nmal Representation of the Curve α E 3. Let N = N (s be the principal nmal vect field of the curve s α(s. The spherical curve α N = N on S 2 is called II.nd spherical representation f α is called the spherical representation of the principal nmals of α. Let s I be the arc length of α. If we denote the arc length of α N by s N, we may write Meover letting dα N N = T N, we obtain α N (s N = N (s. T N = ( κ T + B N Hence we have N = κ Note that κ is the total curvature function of α. Therefe we reach the following result: s N = κ c in terms of H = κ, s N = 1 + H 2 + c.

3 A NEW CHARACTERIZATION FOR INCLINED CURVES 73 Thus we have the following theem: Theem 2.2. α E 3 is an dinary helix if and only if s N = 1 + H 2 s + c The Arc Length of Binmal Representation of the Curve α E 3. Let B = B (s be the binmal vect field of the curve s α(s. The spherical curve α B = B on S 2 is called III.rd spherical representation f α is called the spherical representation of the binmals of α. Let s I be the arc length parameter of α. If we denote the arc length parameter of α B by s B, we may write α B (s B = B (s. Meover letting dα B B = T B, we obtain T B = N B. Hence we have B and s B = + c in terms of the harmonic curvature of α we obtain s B = Thus we give the following theem: κ + c. H Theem 2.3. α E 3 is an dinary helix if and only if s B = κ H + c The Arc Length of Darboux Spherical Representation of the Curve α E 3. Let w = T + κ B be the Darboux vect field of the curve W W s α(s. Let us define the curve α C = C on S 2 by the help of the vect fiel =.This curve is called IV.th spherical representation of α is called the Darboux representation of α. Let s C be the arc length of α C. Then we have α C = C (sc = W. Let us denote the angle between W and T by ϕ (see Figure 1. W = Figure 1

4 74 H. HILMI HACISALIHOĞLU Hence (1 κ = W sin ϕ and = W cos ϕ. Therefe we may write From this last equality we get Hence we have (2 C = cos ϕ T + sin ϕ B. =. C C = = (cos ϕ T + (sin ϕ B = ( sin ϕ T + cos ϕ B dϕ. = dϕ = C. From this equations, in (1 we obtain κ (3 = tan ϕ. Therefe, differentiating with respect to s we have ( κ = (1 + tan 2 ϕ dϕ ( κ [ ( κ ] 2 dϕ = 1 +. From (3, since we have ( dϕ κ = 1 + ( κ 2 and since we have H = κ, we get dϕ Hence from (2, we obtain hence C = H 1+H 2 implies that Since H = dh = H 1 + H 2. C = H 1 + H 2 C = H 1 + H 2 s C = implies H = dh, H + c. 1 + H2

5 A NEW CHARACTERIZATION FOR INCLINED CURVES 75 then we have s C = Arc tan H + c. Thus we give the following theem: Theem 2.4. The curve α E 3 is an inclined curve if and only if s C = const. References [1] K. Sakomato, Helical immersions into a unit sphere, Math. Ann. 261 ( [2] Y. Hong and C. S. Houh, Helical immersions and nmal sections. Kodai Math. J. 8 (1985, [3] Hayden HA., On a generalized helix in a Riemannian n-space. Proc. London Math. Soc., 12 (1986, [4] J. Monterde, Curves with constant curvature Ratios, arxiv. mat. DG/ (2004. [5] M. C. Romero-Fuster, E. Sanabria-Codesal, Generalized helices, twistings and flattenings of curves in n-space, Mathematica Contempanea, Vol. 17 (1999, [6] Barros M. General helices and a theem of Lancert. Proc. AMS 1997 ; 125 : [7] Chouaieb N, Giely A. Maddocks JH. Helices PNAS 2006 ; 103 (25 : [8] Cook TA. the curves of life, Constable, London-1914 ; Reprinted (Dover, London [9] Hayden HA., On a generalized helix in a Riemannian n-space. Proc. London Math. Soc. ( ; 32 : [10] Bektaş M., Balgetir H. and Ergüt M., On a charactarerization of null helix, Bull. Inst. Math. Acad. Sinica, 29 (2001, [11] Bektaş M., Balgetir H. and Ergüt M., Inclined curves null curves in the 3-dimensional Lentzian manifold and their characterization. J. Inst. Math. Comp. Sci., 12 (1999, [12] Ekmekçi, N. and Ilarslan K., On characterization of general helices in Lentz Space, Hadronic J.,23 (2000,no.6, [13] Ekmekçi, N. and Hacısalihoğlu, H. H., On helices of Lentzian manifol, Comm. Fac.Sci. Univ. Ankara, Series A 1. V. 45, pp (1996. [14] Özdamar, E. and Hacısalihoğlu, H. H. A characterization of inclined curves in Euclidean n-space. Comm. Fac. Sci. Univ. Ankara, Series A 1, V. 24, pp (1975. [15] Arslan K., Çelik Y. and Hacısalihoğlu H. H. On harmonic curvatures of a Frenet curves, Comm. Fac. Sci. Univ. Ankara, Series A 1,V.48, pp 15-23, [16] Ekmekçi, N. and Hacısalihoğlu, H. H., Ilarslan K., Harmonic curvatures in Lentzian space, Bull. Malays. Math. Sci. Soc. (2 23 (2000, no. 2, [17] Hacısalihoğlu H. H. and Öztürk R., On the characterization of inclined curves in En I.Tens,N.S. V. 64 (2003. [18] Hacısalihoğlu H. H. and Öztürk R., On the characterization of inclined curves in En II.Tens,N.S. V. 64 (2003. [19] Ekmekçi, N. and Hacısalihoğlu, H. H., On characterization of general helices of a Lentzian manifold. Comm. Fac. Sci. Univ. Ankara, Series A 1, V. 45, pp (1996. [20] Camci, C., Ilarslan K., Kula, L. and Hacısalihoğlu, H. H., Harmonic curvatures and generalized helices in E n, Chaos Solutions Fractals, V. 40, No:5, pp (2009. Department of Mathematics, Ankara University, Beşevler-Ankara/Turkey address: hacisali@science.ankara.edu.tr