Rolling maps in a Riemannian framework. (English) Zbl 1254.53018

Hüper, Knut; Krakowski, Krzysztof Andrzej; Silva Leite, Fátima

Cardoso, João (ed.) et al., Mathematical papers in honour of Fátima Silva Leite. Selected papers based on the presentations at the special session on geometric control theory – a tribute to Fátima Silva Leite on the occasion of her 60th anniversary, held at the 9th Portuguese conference on automatic control, CONTROLO’2010, Coimbra, Portugal, September 8–10, 2010. Coimbra: Universidade de Coimbra, Departamento de Matemática (ISBN 978-972-8564-47-6/pbk). Textos de Matemática. Série B 43, 15-30 (2011).

The rolling of connected manifolds upon another has been studied before in an Euclidean setting. The contribution of this paper is that both involved manifolds are embedded in a Riemannian manifold. The definition given for such a rolling follows closely the well known definition for rolling Euclidean submanifolds with some obvious adaptations. It is shown that the definition reduces to the classical definition when the embedding space is Euclidean. Furthermore, it is shown that in case of existence the rolling map is unique. Then, the results are applied to rolling of non-Euclidean manifolds resulting form deformations of Euclidean manifolds. As an explicit example, the rolling equations for rolling ellipsoids are given.
For the entire collection see [Zbl 1227.00042].

Reviewer: Manfred Husty (Innsbruck)

MSC:

53A17	Differential geometric aspects in kinematics
53A35	Non-Euclidean differential geometry
70G45	Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
93C15	Control/observation systems governed by ordinary differential equations

Keywords:

rolling map; deformations of Euclidean manifolds; rolling ellipsoids