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..


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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: Conference or Workshop Item (Paper)
Venue - Dates: 1st International Meeting on Geometry and Topology, 1997-01-01
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ePrint ID: 29905
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Date Deposited: 15 May 2007
Last Modified: 16 Apr 2017 22:20
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