Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 43, No. 2, pp. 407-410 (2002)

Osculating Plane Preserving Diffeomorphisms

F. J. Craveiro de Carvalho

Departamento de Matematica, Universidade de Coimbra, Portugal, e-mail:

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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