# The parallel group of a plane curve

de Carvalho, F.J. Craveiro and Robertson, S.A.
(1997)
The parallel group of a plane curve.
In,
do Vale, A. Pereira and Pinto, M.R. (eds.)
*Proceedings of the 1st International Meeting on Geometry and Topology. *
* 1st International Meeting on Geometry and Topology*
Germany,
European Mathematical Information Service.

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## Description/Abstract

For any smooth immersion f of the circle in the plane, the parallel group P(f) consists of all self-diffeomorphisms of the circle such that the normal lines at points of each orbit are parallel. The action of P(f) on S^1 cannot be transitive. Thus, for example, P(f)\neq SO(2). We construct examples where P(f) contains a subgroup isomorphic to the group of self-diffeomorphisms of a closed interval (fixing the end-points), is isomorphic to the cyclic group Z_n for any n\epsilon N, and to the dihedral group D_{n}, for any n\epsilon N. If the curvature of f is nowhere zero, however, then P(f) is cyclic of even order.

Item Type: | Book Section |
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Related URLs: | |

Subjects: | Q Science > QA Mathematics |

Divisions: | University Structure - Pre August 2011 > School of Mathematics > Pure Mathematics |

ePrint ID: | 29905 |

Date Deposited: | 15 May 2007 |

Last Modified: | 27 Mar 2014 18:18 |

URI: | http://eprints.soton.ac.uk/id/eprint/29905 |

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