Generalized helices in three-dimensional Lie groups

Generalized helices in three-dimensional Lie groups

We introduce three types of helices in three-dimensional Lie groups with left-invariant metric and give theirgeometrical description similar to that of Lancret. We generalize the results known for the case of three-dimensional Liegroups with bi-invariant metric.

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  • [1]Barros M. General helices and a theorem of Lancret. Proceedings of the American Mathematical Society 1997; 125 (5): 1503-1509.
  • [2] Ciftci U. A generalization of Lancret’s theorem. Journal of Geometry and Physics 2009; 59: 1597-1603.
  • [3] Dogan F. The detailed proof of theorem which characterizes a slant helix. New Trends in Mathematical Sciences 2016; 4 (2): 56-60.
  • [4] Izumiya S, Takeuchi N. New special curves and developable surfaces. Turkish Journal of Mathematics 2004; 28: 153-163.
  • [5] Lucas P, Ortega-Yagües JA. Slant helices in the three-dimensional sphere. Journal of the Korean Mathematical Society 2017; 54: 1331-1343.
  • [6] Menninger A. Characterization of the slant helix as successor curve of the general helix. International Electronic Journal of Geometry 2014; 7 (2): 84-91.
  • [7] Milnor J. Curvatures of left invariant metrics on Lie Groups. Advances in Mathematics 1976; 21: 293-329.
  • [8] Okuyucu OZ, Gok I, Yayli Y, Ekmekci N. Slant helices in three dimensional Lie groups. Applied Mathematics and Computation 2013; 221: 672-683.
  • [9] Özgör C, Gezgin F. On some curves of AW (k)-type. Differential Geometry - Dynamical Systems 2005; 7: 74-80.