Article (Scientific journals)
Geometric conditions for the existence of a rolling without twisting or slipping
Godoy Molina, Mauricio; Grong, Erlend
2014In Communications on Pure and Applied Analysis, 13 (1), p. 435-452
Peer reviewed
 

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Keywords :
Rolling maps; geodesic curvatures; anti-development
Abstract :
[en] We give a complete answer to the question of when two curves in two different Riemannian manifolds can be seen as trajectories of rolling one manifold on the other without twisting or slipping. We show that, up to technical hypotheses, a rolling along these curves exists if and only if the geodesic curvatures of each curve coincide. By using the anti-developments of the curves, which we claim can be seen as a generalization of the geodesic curvatures, we are able to extend the result to arbitrary absolutely continuous curves. For a manifold of constant sectional curvature rolling on itself, two such curves can only differ by an isometry. In the case of surfaces, we give conditions for when loops in the manifolds lift to loops in the configuration space of the rolling.
Disciplines :
Mathematics
Author, co-author :
Godoy Molina, Mauricio;  University of Bergen > Department of Mathematics
Grong, Erlend ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit
Language :
English
Title :
Geometric conditions for the existence of a rolling without twisting or slipping
Publication date :
2014
Journal title :
Communications on Pure and Applied Analysis
Volume :
13
Issue :
1
Pages :
435-452
Peer reviewed :
Peer reviewed
Funders :
The first author is partially supported by an ``Emmy-Noether scholarship'' of the DFG. The second author is partially supported by the Fonds National de la Recherche Luxembourg.
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since 30 September 2014

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