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The parallel group of a plane curve

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

Record type: Conference or Workshop Item (Paper)


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.

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Published date: 1997
Venue - Dates: 1st International Meeting on Geometry and Topology, 1997-01-01


Local EPrints ID: 29905
PURE UUID: a37bf3f7-bdaf-4c60-a243-50b2e24c2336

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Date deposited: 15 May 2007
Last modified: 17 Jul 2017 15:56

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Author: F.J. Craveiro de Carvalho
Author: S.A. Robertson
Editor: A. Pereira do Vale
Editor: M.R. Pinto

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