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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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
Accepted Date and Publication Date:
Date Deposited: 15 May 2007
Last Modified: 31 Mar 2016 11:56

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