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 selfdiffeomorphisms 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 selfdiffeomorphisms of a closed interval (fixing the endpoints), 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  
URI:  http://eprints.soton.ac.uk/id/eprint/29905 
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