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
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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