Legendre curves and the singularities of ruled surfaces obtained by using rotation minimizing frame
Abstract
UDC 514.7
In this paper, Legendre curves in unit tangent bundle are given using rotation minimizing vector fields. Ruled surfaces corresponding to these curves are represented. Singularities of these ruled surfaces are also analyzed and classified.
References
S. C. Anco, Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces, J. Geom. and Phys., 58, 1 – 37 (2008), https://doi.org/10.1016/j.geomphys.2007.09.005 DOI: https://doi.org/10.1016/j.geomphys.2007.09.005
C. Baikoussis, D. E. Blair, On Legendre curves in contact $3$-manifolds, Geom. Dedicata, 49, № 2, 135 – 142 (1994), https://doi.org/10.1007/BF01610616 DOI: https://doi.org/10.1007/BF01610616
U. Beyhan, I. G¨ok, Y. Yayli, A new approach on curves of constant precession, Appl. Math. and Comput., 27, 317 – 323 (2016), https://doi.org/10.1016/j.amc.2015.11.083 DOI: https://doi.org/10.1016/j.amc.2015.11.083
M. Bekar, Y. Yayli, Slant helix curves and acceleration centers in Minkowski 3-space $R^3_1$, J. Adv. Phys., 6, 133 – 141 (2017). DOI: https://doi.org/10.1166/jap.2017.1306
R. L. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly, 82, 246 – 251 (1975), https://doi.org/10.2307/2319846 DOI: https://doi.org/10.1080/00029890.1975.11993807
J. W. Bruce, P. J. Giblin, Curves and singularities, 2nd. ed., Cambridge Univ. Press, Cambridge (1992), https://doi.org/10.1017/CBO9781139172615 DOI: https://doi.org/10.1017/CBO9781139172615
F. Etayo, Rotation minimizing vector fields and frames in Riemannian manifold, Proc. Math. and Statist., 161, 91 – 100 (2016), https://doi.org/10.1007/978-3-319-32085-4_8 DOI: https://doi.org/10.1007/978-3-319-32085-4_8
R. T. Farouki, Pythagorean-hodograph curves: algebra and geometry inseparable, Geom. and Comput., 1, Springer, Berlin (2008), https://doi.org/10.1007/978-3-540-73398-0 DOI: https://doi.org/10.1007/978-3-540-73398-0
L. Haiming, P. Donghe, Legendrian dualities between spherical indicatrixes of curves and surfaces according to Bishop frame, J. Nonlinear Sci. and Appl., № 5, 1 – 13 (2016), https://doi.org/10.22436/jnsa.009.05.82 DOI: https://doi.org/10.22436/jnsa.009.05.82
F. Hathout, M. Bekar, Y. Yayli, $N$-Legendre and $N$-slant curves in the unit tangent bundle of surfaces, Kuwait J. Sci., 44, № 3, 106 – 111 (2017).
F. Hathout, M. Bekar, Y. Yayli, Ruled surfaces and tangent bundle of unit 2-sphere, Int. J. Geom. Methods Mod. Phys., 14, № 10, Article 1750145 (2017), https://doi.org/10.1142/S0219887817501456 DOI: https://doi.org/10.1142/S0219887817501456
S. Izumiya, N. Takeuchi, New special curves and developable surfaces, Turkish J. Math., 28, № 2, 153 – 163 (2004).
G. Mari Beffa, Poisson brackets associated to invariant evolutions of Riemannian curves, Pacif. J. Math., 125, 357 – 380 (2004), https://doi.org/10.2140/pjm.2004.215.357 DOI: https://doi.org/10.2140/pjm.2004.215.357
O. P. Shcherbak, Projectively dual space curve and Legendre singularities, Sel. Math. Sov., 5, 391 – 421 (1986).
Y. Tashiro, On contact structure of hypersurfaces in complex manifolds, Tohoku Math. J., 15, 62 – 78 (1963), https://doi.org/10.2748/tmj/1178243870 DOI: https://doi.org/10.2748/tmj/1178243870
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