Abstract: For someone familiar with the notion of self-parallel group for an immersion into euclidean space [FR], [W] it is only natural to wonder what happens if, in the case of space curves, normal planes are replaced by, say, osculating planes. We give here a necessary and sufficient condition for the non-triviality of the osculating group of a simple space curve. This is, in the new situation, the group corresponding to the self-parallel group. Problems of a similar nature have also been considered in [BCW] and [CR].
[BCW] d'Azevedo Breda, A. M.; Craveiro de Carvalho, F. J.; Wegner, Bernd: On the existence of Bertrand pairs. Pré-publiccao 01-06, Departamento de Matematica da Universidade de Coimbra.
[CR] Craveiro de Carvalho, F. J.; Robertson, S. A.: The parallel group of a plane curve. Proceedings of the 1$^{st}$ International Meeting on Geometry and Topology, Braga (1997), 57-61.
[FR] Farran, H. R.; Robertson, S. A.: Parallel immersions in euclidean space. J. London Math. Soc. 35 (1987), 527-538.
[W] Wegner, B.: Self-parallel and transnormal curves. Geom. Dedicata 38 (1991), 175-191.
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