Self-parallel and transnormal curves (Q911054)

This is an extension of the paper by \textit{F. J. Craveiro de Carvalho} and \textit{S. A. Robertson} in Math. Scand. 65, 67-74 (1989; see the preceding review). The more geometric interpretation of the parallelism of immersions through parallel sections of the normal bundle [see the author's paper in J. Differ. Geom. 16, 93-100 (1981; Zbl 0406.53005) or `Some remarks on parallel immersions' (Colloq. Math. Soc. Janos Bolyai (to appear 1990; Zbl 0687.53051)] is shown to give a better insight into the structure of self-parallel curves. Using this tool, the construction of examples from a closed central curve, given in the first quoted paper, can be extended to general closed regular curves without assuming nonvanishing curvature. Furthermore every self-parallel curve in 3-space is shown to be of the type obtained by this construction, if the order of the self-parallel group is greater than 2. These considerations yield a new possibility to construct transnormal curves. This is used to present examples for transnormal curves in 4- space with arbitrarily high degree of transnormality, disproving a long- standing conjecture of \textit{M. C. Irwin} [J. Lond. Math. Soc. 42, 545-552 (1967; Zbl 0152.223)]. Also injective infinitely transnormal imbeddings of the real line into 4-space are shown to exist. They do not exist in 3- space. All results on self-parallelism of curves can be transferred to piecewise linear curves in Euclidean spaces [see the author, `Exterior parallelism for polyhedra', Mathematica Pannonica (to appear 1990)] giving good facilities for drawing pictures of these curves.