A New Representation for Slant Curves in Sasakian 3-Manifolds

Yıl 2024, , 277 - 289, 23.04.2024

Osman Ateş , İsmail Gök , Yusuf Yaylı

https://doi.org/10.36890/iejg.1397659

Öz

In this paper, we define a new kind of curve called $N$-slant curve whose principal normal vector field makes a constant angle with the Reeb vector field $\xi$ in Sasakian $3$-manifolds. Then, we give some characterizations of $N$-slant curves in Sasakian $3$-manifolds and we obtain some properties of the curves in $\mathbb{R}^{3}(-3)$. Moreover, we investigate the conditions of $C$-parallel and $C$-proper mean curvature vector fields along $N$-slant curves in Sasakian $3$-manifolds. Finally, we study $N$-slant curves of type $AW(k)$ where k=1,2 or 3.

Anahtar Kelimeler

Slant helices, Mean curvature vector fields, Curves of AW(k)-type, Sasakian manifolds

Kaynakça

• [1] Ahmad, T.A, Turgut, M.: Some characterizations of slant helices in the Euclidean space En . Hacettepe Journal of Mathematics and Statistics. 39, 327–336 (2010).
• [2] Altunkaya, B.: Slant Helices that Constructed from Hyperspherical Curves in the n-dimensional Euclidean Space. International Electronic Journal of Geometry. 12(2), 229–240 (2019).
• [3] Arslan, K., West, A.: Product submanifols with pointwise 3-Planar normal sections. Glasgow Math. J. 37, 73–81 (1995).
• [4] Arslan, K., Özgür, C.: Curves and surfaces of AW(k) typ. Geometry and Topology of Submanifolds, IX (Valenciennes/Lyon/Leuven, 1997), World Sci. Publishing, River Edge, NJ, 21-26 (1999). https://doi.org/10.1142/9789812817976-0003.
• [5] Baikoussis, C., Blair, D.E.: On Legendre curves in contact 3-manifolds. Geom. Dedicate, 49, 135-142 (1994). https://doi.org/10.1007/BF01610616
• [6] Barros, M.: General Helices and a theorem of Lancert. Proc. Amer. Math. Soc., 125(5), 1503-1509 (1997).
• [7] Blair, D.E.: Contact manifolds in Riemannian geometry. Lecture Notes in Math. 509, Springer, Berlin, Hiedelberg, New York, (1976).
• [8] Blair, D.E.: Riemannian geometry of contact and simplectic manifolds. Birkhauser, Boston, (2002).
• [9] Camcı, Ç.: Extended cross product in a 3- dimensional almost contact metric manifold with applications to curve theory. Turk. J. Math., 35, 1-14 (2011). https://doi.org/10.3906/mat-0910-103
• [10] Chen, B.Y.: Total Mean curvature and submanifolds of finite type. Series in Pure Mathematics, 1, World Scientific Publishing Co., Singapore, (1984). https://doi.org/10.1142/9237
• [11] Cho, J.T., Inoguchi, J.-I., Lee, J.E.: On slant curves in Sasakian 3-manifolds. Bull. Austral. Math. Soc., 74, 359-367 (2006). https://doi.org/10.1017/S0004972700040429
• [12] Cho, J.T., Inoguchi, J.-I., Lee, J.E.: Biharmonic curves in 3-dimensional Sasakian space forms. Ann. Mat. Pura Appl., 186, 685-701 (2007). https://doi.org/10.1007/s10231-006-0026-x
• [13] Cho, J.T., Lee, J.E.: Slant curves in contact Pseudo-Hermitian 3-manifolds. Bull. Austral. Math. Soc., 78, 383-396 (2008). https://doi.org/10.1017/S0004972708000737
• [14] Inoguchi, J.-I., Lee, J.E.: On slant curves in normal almost contact metric 3-manifolds. Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry, 55, 603-620 (2014).
• [15] Inoguchi, J.-I., Lee, J.E.: Slant curves in 3-dimensional almost contact metric geometry. International Electronic Journal of Geometry, 8(2), 106-146 (2015).
• [16] Izumiya, S., Takeuchi, N.: New special curves and developable surfaces. Turk J. Math., 28, 153-163 (2004).
• [17] Kula, L., Yaylı, Y.: On slant helix and its spherical indicatrix. Applied Mathematics and Computation 169, 600-607 (2005). https://doi.org/10.1016/j.amc.2004.09.078
• [18] Lancret, M.A.: Mémoire sur les courbes à double courbure. Mémoires présentés à l’Institut1", 416-454 (1806).
• [19] Lee, C. W., Lee, J. W.: Classifications of special curves in the Three-Dimensional Lie Group. International Journal of Mathematical Analysis, 10(11), 503-514 (2016).
• [20] Lee, J.E., Suh Y.J., Lee, H.: C-parallel mean curvature vector fields along slant curves Sasakian 3-manifolds. Kyungpook Math. J., 52(1), 49-59 (2012). https://doi.org/10.5666/KMJ.2012.52.1.49
• [21] Okuyucu, O.Z., Gök, ˙I., Yaylı,Y., Ekmekci, F.N.: Slant helices in three dimensional Lie groups. Applied Mathematics and Computation, 221, 672–683 (2013). https://doi.org/10.1016/j.amc.2013.07.008
• [22] Olszak, Z.: Normal almost contact metric manifolds of dimension three. Annales Polonici Mathematici, 47, 41-50 (1986).
• [23] Özgür, C., Gezgin F.: On some curves of AW (k)-type. Differ. Geom. Dyn. Syst, 7, 74-80 (2005).
• [24] Özgür, C., Tripathi, M.M.: On Legendre curves in α − Sasakian manifolds. Bull. Malays. Math. Sci. Soc. (2), 31(1), 91-96 (2008).
• [25] Özgür, C., Güvenç, ¸S: On some types of slant curves in contact pseudo-Hermitian 3-manifolds. Ann. Polon. Math. 104, 217-228 (2012), https://doi.org/10.4064/ap104-3-1.
• [26] Özgür, C., Güvenç, ¸S. On some classes of curves in contact pseudo-Hermitian 3-manifolds. Riemannian Geometry and Applications, RIGA 2011 Ed. Univ. Bucure¸sti, Bucharest, 229–238 (2011).
• [27] Simons, J.: Minimal varieties in Riemannian manifolds. Ann. of Math., 88(2), 62-105 (1968). https://doi.org/10.2307/1970556
• [28] Struik, D.J.: Lectures on Classical Differential Geometry. Dover, New-York, (1988).
• [29] Yaylı, Y., Zıplar, E. On slant helices and general helices in Euclidean n-space. Mathematica Aeterna 1(8), 599-610 (2011).
• [30] Yıldırım, A., On curves in 3-dimensional normal almost contact metric manifolds. Int. J. Geom. Methods M., 18(1), 2150004, (2021), https://doi.org/10.1142/S0219887821500043
• [31] Yoon, D.W.: General helices of AW (k)-type in the Lie group. Journal of Applied Mathematics, (2012), https://doi.org/10.1155/2012/535123.