Some New Properties of Surfaces Generated by Null Cartan Curves

Year 2022, , 116 - 131, 30.04.2022

Ayşe Yavuz , Melek Erdoğdu

https://doi.org/10.36890/iejg.963159

Abstract

In this paper, some special types of surfaces with null Cartan base curve are investigated. The generating lines of the surfaces are chosen as a linear combination of Cartan frame fields with non-constant differentiable functions. Firstly, the surfaces whose generating lines have the same direction of Cartan frame fields B; N and T are examined respectively. As a special case, Gaussian and Mean curvatures of one parameter family of Bertrand curves of a given null Cartan curve and the singular points of this type of surface are stated. Furthermore, an example is also stated to explain the obtained results. Then, the surfaces with null Cartan base curve are investigated where generating lines lie on the planes spanned by (N, B), (T,B) and (T, N) respectively. Finally, some differential geometric properties of these surface are given mainly in three different cases

Keywords

Cartan frame, Bertrand curve, null Cartan curve

References

• [1] Akamine, S.: Behavior of the Gaussian curvature of timelike minimal surfaces with singularities. Hokkaido Mathematical Journal. 48 (3), 537-568 (2019). https://doi.org/10.14492/hokmj/1573722017
• [2] Aslan, S., Bekar, M., Yaylı, Y.: Ruled surfaces constructed by quaternions. Journal of Geometry and Physics. 161, (2021). https://doi.org/10.1016/j.geomphys.2020.104048
• [3] Clelland, J.N.: Totally quasi-umbilic timelike surfaces in R21. Asian J. Math. 16(1), 189-208 (2012).
• [4] Erdo˘gdu, M., Yavuz, A.: On Backlund Transformation and Motion of Null Cartan Curves.International Journal of Geometric Methods in Modern Physics. 19 (1), (2022). https://doi.org/10.1142/S0219887822500141
• [5] Honda, K., Inoguchi, J.I.: Deformation of Cartan framed null curves preserving the torsion. Differ. Geom. Dyn. Syst. 5 (1), 31-37 (2003).
• [6] Kim, Y.H., Yoon, D.W.: Classification of ruled surfaces in Minkowski 3-spaces. Journal of Geometry and Physics. 49 (1), 89-100 (2004). https://doi.org/10.1016/S0393-0440(03)00084-6
• [7] Kim, Y.H., Yoon, D.W.: On non-developable ruled surfaces in Lorentz-Minkowski 3-spaces. Taiwanese Journal of Mathematics. 11 (1), 197-214 (2007). https://doi.org/10.11650/twjm/1500404646
• [8] Liu, H.: Characterizations of ruled surfaces with lightlike ruling in Minkowski 3-space. Results in Mathematics. 56 (1), (2009). https://doi.org/10.1007/s00025-009-0431-8
• [9] Nolasco, B., Pacheco, R.: Evolutes of plane curves and null curves in Minkowski 3-space. Journal of Geometry. 108 (1), 195-214 (2017). https://doi.org/10.1007/s00022-016-0334-2
• [10] Wang, Z., Pei, D.: Singularities of ruled null surfaces of the principal normal indicatrix to a null Cartan curve in de Sitter 3-space. Physics Letters B. 689 (2-3), 101-106 (2010). https://doi.org/10.1016/j.physletb.2010.04.050
• [11] Yaylı, Y.: On the motion of the Frenet vectors and spacelike ruled surfaces in the Minkowski 3-space. Mathematical and Computational Applications. 5 (1), 49-55 (2000).
• [12] Yüca, G.: Kinematics applications of dual transformations. Journal of Geometry and Physics. 163, (2021). https://doi.org/10.1016/j.geomphys.2021.104139