• Kisi, Ilim (Department of Mathematics, Kocaeli University) ;
  • Ozturk, Gunay (Department of Mathematics, Ismir Democracy University)
  • Received : 2017.12.29
  • Accepted : 2018.02.09
  • Published : 2018.06.25


In the present study, we consider the curves in Galilean 4-space ${\mathbb{G}}_4$. We find out the involute curves of order k (k = 1, 2, 3) of a given curve. We get the relationships between the Frenet apparatus of a given curve and its involute curves of order k.


  1. A. Z. Azak, M. Akyigit and S. Ersoy, Involute-evolute curves in Galilean space ${\mathbb{G}}_3$, Sci. Magna 4 (2010), 75-80.
  2. B. Divjak and Z. M. Sipus, Involutes and evolutes in n-dimensional simply isotropic space, Journal of Information and Organizational Sciences 23 (1999), 71-79.
  3. M. Elzawy and S. Mosa, Smarandache curves in the Galilean 4-space ${\mathbb{G}}_4$, Journal of the Egyptian Mathematical Society, 25 (2017), 53-56.
  4. H. Gluck, Higher curvatures of curves in Euclidean space, Am. Math. Monthly 73 (1966), 699-704.
  5. G.P. Henderson, Parallel curves, Canad. J. Math. 6 (1954), 99-107.
  6. O.B. Kalkan, Position vector of a W-curve in the 4D Galilean space ${\mathbb{G}}_4$, Facta Universitatis Ser. Math. Inform. 31 (2016), 485-492.
  7. B. Kilic, K. Arslan and G. Ozturk, Tangentially cubic curves in Euclidean spaces, Differential Geometry-Dynamical Systems 10 (2008), 186-196.
  8. F. Klein and S. Lie, Uber diejenigen ebenenen kurven welche durch ein geschlossenes system von einfach unendlich vielen vartauschbaren linearen Transformationen in sich ubergehen, Math. Ann. 4 (1871), 50-84.
  9. M. S. Lone, Some characterisations of rectifying curves in four dimensional Galilean space ${\mathbb{G}}_4$, Global Journal of Pure and Applied Mathematics 13 (2017), 579-587.
  10. J. Monterde, Curves with constant curvature ratios, Bull. Mexican Math. Soc. Ser. 13 (2007), 177-186.
  11. H. Oztekin, Special Bertrand curves in 4D Galilean space, Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 318458, 7 pages.
  12. G. Ozturk, K. Arslan and H. H. Hacisalihoglu, A characterization of ccr-curves in ${\mathbb{R}}^m$, Proc. Estonian Acad. Sci. 57 (2008), 217-224.
  13. G. Ozturk, On involutes of order k of a space-like curve in Minkowski 4-space ${\mathbb{E}}^4_1$, AKU J. Sci. Eng. 16 (2016), 569-575.
  14. G. Ozturk, K. Arslan and B. Bulca, A characterization of involutes and evolutes of a given curve in ${\mathbb{E}}^n$, Kyungpook Mathematical Journal 58 (2018), 117-135.
  15. G. Ozturk, S. Buyukkutuk and I. Kisi, A characterization of curves in Galilean 4-space ${\mathbb{G}}_4$, Bull. Iranian Math. Soc. 43 (2017), 771-780.
  16. O. Roschel, Die Geometrie Des Galileischen Raumes, Forschungszentrum Graz Research Centre, Austria, 1986.
  17. I. M. Yaglom, A Simple Non-Euclidean Geometry and Its Physical Basis, Springer-Verlag Inc., New York, 1979.
  18. S. Yilmaz, Construction of the Frenet-Serret frame of a curve in 4D Galilean space and some applications, International Journal of the Physical Sciences Vol. 5 (2010), 1284-1289.