19027

On the Framed Normal Curves in Euclidean 4-space

In this paper, we introduce the adapted frame of framed curves and we give the relations between the adapted frame and Frenet type frame of the framed curve in four-dimensional Euclidean space. Moreover, we define the framed normal curves in four-dimensional Euclidean space. We obtain some characterizations of framed normal curves in terms of their framed curvature functions. Furthermore, we give the necessary and sufficient condition for a framed curve to be a framed normal curve.

Keywords:

Framed curves Normal curves, Singular points,

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[1] S. Honda, M. Takahashi, Framed curves in the Euclidean space, Adv. Geom., 16 (2016), 265-276.
[2] Y. Wang, D. Pei, R. Gao, Generic properties of framed rectifying curves, Mathematics, 7(1) (2019), 37.
[3] A. Kızılay, Ö G. Yıldız, O. Z. Okuyucu, Evolution of quaternionic curve in the semi-Euclidean space E4 2 , Math. Meth. Appl. Sci., 44 (2021), 7577–7587, https://doi.org/10.1002/mma.6374.
[4] B. Dog˘an Yazıcı, S. Özkaldı Karakuş, M. Tosun, Framed normal curves in Euclidean space, Tbilisi Math. J., (2020), 27-37.
[5] B. Dog˘an Yazıcı, S. Özkaldı Karakuş, M. Tosun, On the classification of framed rectifying curves in Euclidean space, Math. Meth. Appl. Sci., (2021), 1-10, https://doi.org/10.1002/mma.7561.
[6] C. Özyurt, Singular curves and their properties, Master Thesis, Ankara University, 2021.
[7] S. Honda, T. Masatomo, Evolutes of framed immersions in the Euclidean space, Hokkaido Uni. Preprint Ser. Math., 1095 (2016), 1-24.
[8] T. Fukunaga, M. Takahashi, Existence and uniqueness for Legendre curves, J. Geom. 104 (2013), 297-307.
[9] K. Eren, Ö . G. Yıldız, M. Akyiğit, Tubular surfaces associated with framed base curves in Euclidean 3-space, Math. Meth. Appl. Sci., (2021), 1- 9, https://doi.org/10.1002/mma.7590.
[10] B. Y. Chen, When does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Monthly, 110 (2003), 147-152.
[11] K. Ilarslan, E. Nesovic, Timelike and null normal curves in Minkowski space E3_1 , Indian J. Pure Appl. Math., 35(7) (2004), 881-888.
[12] K. Ilarslan, Spacelike Normal Curves in Minkowski Space E3_1 , Turk. J. Math., 29 (2005), 53-63.
[13] K. Ilarslan, E. Nesovic, Spacelike and timelike normal curves in Minkowski space-time, Publ. Inst. Math. Belgrade, 85(99) (2009), 111-118.
[14] F. Gökçelik, Z. Bozkurt, ˙I. Gök, F. N. Ekmekçi, Y. Yaylı Parallel transport frame in 4-dimensional Euclidean space, Caspian J. Math. Sci., 3 (2014), 91- 103.