The parallel group of a plane curve
The parallel group of a plane curve
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.
European Mathematical Information Service
de Carvalho, F.J. Craveiro
7f55d2e3-677c-4955-9fef-f81497035bb2
Robertson, S.A.
95eb6b84-69cb-407a-8e43-12a12534cfdc
1997
de Carvalho, F.J. Craveiro
7f55d2e3-677c-4955-9fef-f81497035bb2
Robertson, S.A.
95eb6b84-69cb-407a-8e43-12a12534cfdc
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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Conference or Workshop Item
(Paper)
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.
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Published date: 1997
Venue - Dates:
1st International Meeting on Geometry and Topology, Braga, Portugal, 1997-01-01
Identifiers
Local EPrints ID: 29905
URI: http://eprints.soton.ac.uk/id/eprint/29905
PURE UUID: a37bf3f7-bdaf-4c60-a243-50b2e24c2336
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Date deposited: 15 May 2007
Last modified: 11 Dec 2021 15:16
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Contributors
Author:
F.J. Craveiro de Carvalho
Author:
S.A. Robertson
Editor:
A. Pereira do Vale
Editor:
M.R. Pinto
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