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AW(k)-type Salkowski Curves in Euclidean 3-Space IE^3

We deal with AW(k)-type Salkowski (anti-Salkowski) curves with constant in the Euclidean 3-space. We show that there is no AW(1)-type Salkowski curve and AW(1)-type anti-Salkowski curve in . Also, we handle weak AW(2)-type and weak AW(3)-type Salkowski (anti-Salkowski) curves. Also, we show that there is no weak AW(2)-type Salkowski curve in .

Keywords:

AW(k)-type Salkowski Curve,

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[1] K. Arslan and A. West, “Product submanifols with pointwise 3-planar normal sections,” Glasgow Math. J., vol.37, pp. 73-81, 1995.
[2] K. Arslan and C. Özgür, “Curves and surfaces of AW(k) type,” Geometry and topology of Submanifolds IX, World Scientific, 21-26, 1997.
[3] C. Özgür and F. Gezgin,.”On some curves of AW(k)-type,”. Differential Geometry-Dynamical Systems, vol. 7, pp. 74.80, 2005.
[4] B. Kılıç and K. Arslan, “On curves and surfaces of AW(k)-type,” BAÜ Fen Bil. Enst. Dergisi, vol. 6, no. 1, pp. 52-61, 2004.
[5] I. Kişi, S. Büyükkütük, Deepmala, and G. Öztürk, “AW(k)-type curves according to parallel transport frame in Euclidean space ,” Facta Universitaties, Series: Mathematics and Informatics, vol. 31, no. 4, pp. 885-905, 2016.
[6] I. Kişi and G. Öztürk, “AW(k)-type curves according to the Bishop frame,” arXiv:1305.3381v1 [math.DG], 2013.
[7] D. W. Yoon, “General helices of Aw(k)-type in the Lie group,” J. Appl. Math., pp.1-10, 2012.
[8] N. Chouaieb, A. Goriely, and J.H. Maddocks, “Helices,” PANS, 103, 9398.9403, 2006.
[9] A. A. Lucas and P. Lambin, “Diffraction by DNA, carbon nanotubes and other helical nanostructures,” Rep. Prog. Phys., vol. 68, 1181.1249, 2005.
[10] C. D. Toledo-Suarez, “On the arithmetic of fractal dimension using hyperhelices,” Chaos Solitons and Fractals, vol. 39, pp. 342.349, 2009.
[11] X. Yang, “High accuracy approximation of helices by quintic curve,” Comput. Aided Geometric Design, vol. 20, pp. 303.317, 2003.
[12] A. Gray, “Modern differential geometry of curves and surface,” CRS Press, Inc., 1993.
[13] H. Gluck, “Higher curvatures of curves in Euclidean space,” Amer. Math. Monthly, vol. 73, pp. 699-704, 1966.
[14] F. Klein and S. Lie, “Uber diejenigen ebenenen kurven welche durch ein geschlossenes system von einfach unendlich vielen vartauschbaren lin-earen transformationen in sich übergehen,” Math. Ann,. vol. 4, pp. 50-84, 1871.
[15] J. Monterde, “Curves with constant curvature ratios,” Bulletin of Mexican Mathematic Society,” vol. 13, pp. 177-186, 2007.
[16] G. Öztürk, K. Arslan, and H.H. Hacisalihoğlu, “A characterization of ccr-curves in ,” Proc. Estonian Acad. Sciences, vol. 57, pp. 217-224, 2008.
[17] E. Salkowski, “Zur transformation von raumkurven,” Math. Ann., vol. 66, pp. 517-557, 1909.
[18] A.T. Ali, “Spacelike Salkowski and anti-Salkowski curves with spacelike principal normal in Minkowski 3-space,” Int. J. Open Problems Comp. Math., vol. 2, pp. 451.460, 2009.
[19] A.T. Ali, “Timelike Salkowski and anti-Salkowski curves in Minkowski 3-space,” J. Adv. Res. Dyn. Cont. Syst., vol. 2, pp. 17.26, 2010.
[20] F. Kaymaz and F. K. Aksoyak, “Some special curves and Manheim curves in three dimensional Euclidean space,” Mathematical Sciences and applications E-Notes, vol. 5, no. 1, pp. 34-39, 2017.
[21] J. Monterde, “Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion,” Computer Aided Geometric Design, vol. 26, no. 3, pp. 271-278, 2009.
[22] M. P. Do Carmo, “Differential geometry of curves and surfaces”, PrenticeHall, Englewood Cliffs, N. J. 1976.