A New Characterization of Tzitzeica Curves in Euclidean 4-Space

Year 2023, , 77 - 86, 31.07.2023

Emrah Tunç , Bengü Bayram

https://doi.org/10.54974/fcmathsci.1176710

Cited By: 1

Abstract

In this study, we are interested in Tzitzeica curves (Tz-curves) in Euclidean 4 -space. Tz-curve condition for Euclidean 4 -space are determined as three types for three hyperplanes and some examples are given.

Keywords

Tzitzeica condition, Tzitzeica curve, hyperplane, Frenet frame

References

• Bobe A., Boskoff W.G., Ciuca M.G., Tzitzeica type centro-affine invariants in Minkowski space, Analele Stiintifice ale Universitatii Ovidius Constanta, 20(2), 27-34, 2012.
• Bulca B., A Characterization of Surfaces in E4 , Ph.D., Uludağ University, Bursa, Türkiye, 2012.
• Crasmareanu M., Cylindrical Tzitzeica curves implies forced harmonic oscillators, Balkan Journal of Geometry and Its Applications, 7(1), 37-42, 2002.
• Gray A., Modern Differential Geometry of Curves and Surfaces, CRC Press, 1993.
• Gluck H., Higher curvatures of curves in Euclidean space, The American Mathematical Monthly, 73(7), 243-245, 1966.
• Karacan M.K., Bükcü B., On the hyperbolic cylindrical Tzitzeica curves in Minkowski 3-space, Journal of Balıkesir University Institute of Science and Technology, 10(1), 46-51, 2009.
• Karacan M.K., Bükcü B., On the elliptic cylindrical Tzitzeica curves in Minkowski 3-space, Scientia Magna, 5(3), 44-48, 2009.
• Klein F., Lie S., Uber diejenigen ebenenen kurven welche durch ein geschlossenes system von einfach unendlich vielen vartauschbaren linearen transformationen in sich übergehen, The American Mathematical Monthly, 4, 50-84, 1871.
• Öztürk G., Arslan K., Hacısalihoğlu H., A characterization of ccr-curves in Rn , Proceedings of the Estonian Academy of Sciences, 57, 217-224, 2008.
• Tunç E., A Characterization of Tzitzeica Curves and Surfaces, Ph.D., Balıkesir University, Balıkesir, Türkiye, 2021.
• Tzitzeica G., Sur une nouvelle classe de surfaces, Comptes Rendus des Seances de l’Academie des Sciences Paris, 144(1), 1257-1259, 1907.
• Tzitzeica G., Sur certaines courbes gauches, Annales scientifiques de l’École normale supérieure, 28(3), 9-32, 1911.