A characterization of curves in Galilean 4-space $G_4$

Document Type : Research Paper


Kocaeli University‎, ‎Art and Science Faculty‎, ‎Department of Mathematics‎, ‎Kocaeli‎, ‎Turkey.


‎In the present study‎, ‎we consider a regular curve in Galilean‎ ‎$4$-space $\mathbb{G}_{4}$ whose position vector is written as a‎ ‎linear combination of its Frenet vectors‎. ‎We characterize such‎ ‎curves in terms of their curvature functions‎. ‎Further‎, ‎we obtain‎ ‎some results of rectifying‎, ‎constant ratio‎, ‎$T$-constant and‎ ‎$N$-constant curves in $\mathbb{G}_{4}$‎.


Main Subjects

T.A. Ahmad, Position vectors of curves in Galilean space G3, Mat. Vesnik 64 (2012), no. 3, 200--210.
A.Z. Azak, M. Akyiǧit and S. Ersoy, Involute-Evolute curves in Galilean space G3, Sci. Magna 4 (2010), no. 6, 75--80.
B.Y. Chen, Constant ratio hypersurfaces, Soochow J. Math. 28 (2001), no. 4, 353--362.
 B.Y. Chen, Geometry of warped products as Riemannian submanifolds and related problems, Soochow J. Math. 28 (2002), no. 2, 125--156.
B.Y. Chen, When does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Monthly 110 (2003), no. 2, 147--152.
B.Y. Chen and F. Dillen, Rectifying curves as centrodes and extremal curves, Bull. Inst. Math. Acad. Sin. 33 (2005), no. 2, 77--90.
A. Gray, Modern Differential Geometry of Curves and Surfaces, CRC Press, 1993.
S. Gürpinar, K. Arslan and G. Öztürk, A characterization of constant-ratio curves in Euclidean 3-space E3, Acta Univ. Apulensis Math. Inform. 44 (2015) 39--51.
F. Klein and S. Lie, Uber diejenigen ebenenen kurven welche durch ein geschlossenes system von einfach unendlich vielen vartauschbaren linearen Transformationen in sich
ubergehen. acte, Math. Ann. 4 (1871) 50--84.
J. Monterde, Curves with constant curvature ratios, Bol. Soc. Mat. Mex. 13 (2007), no. 1, 177--186.
A.O. Oǧrenmis, M. Ergüt and M. Bektaş, On the helices in the Galilean space G3, Iran. J. Sci. Technol. Trans. A Sci. 31 (2007), no. 2, 177--181.
A.O. Oĝrenmis, H. Öztekin and M. Ergüt, Bertrand curves in Galilean space and their characterizations, Kragujevac J. Math 32 (2009) 139--147.
H. Öztekin, Special Bertrand curves in 4D Galilean space, Math. Probl. Eng. 2014 (2014), Article ID 318458, 7 pages.
H. Öztekin and A.O. Öǧrenmis, Normal and rectifying curves in pseudo-Galilean space G1 3 and their characterizations, J. Math. Comput. Sci. 2 (2012), no. 1, 91--100.
H.B. Öztekin and S. Tatlıpınar, On some curves in Galilean plane and 3-dimensional Galilean space, J. Dyn. Syst. Geom. Theor. 10 (2012), no. 2, 189--196.
G.  Öztürk, K. Arslan and H.H. Hacisalihoglu, A characterization of ccr-curves in Rn, Proc. Est. Acad. Sci. 57 (2008), no. 4, 217--224.
O. Röschel, Die Geometrie des Galileischen Raumes, Forschungszentrum Graz Research Centre, Austria, 1986.
S. Yilmaz, Construction of the Frenet-Serret frame of a curve in 4D Galilean space and some applications, Int. J. Phys. Sci. 5 (2010), no. 5, 1284--1289.
D.W. Yoon, J.W. Lee and C.W. Lee, Osculating curves in the Galilean 4-Space, Int. J. Pure Appl. Math. 100 (2015), no. 4, 497--506.