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A unifying approach for rolling symmetric spaces

  • * Corresponding author: fleite@mat.uc.pt

    * Corresponding author: fleite@mat.uc.pt

The second and third authors acknowledge Fundaç˜ao para a Ciência e Tecnologia (FCT) and COMPETE 2020 program for the financial support to the project UIDB/00048/2020

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  • The main goal of this paper is to present a unifying theory to describe the pure rolling motions of Riemannian symmetric spaces, which are submanifolds of Euclidean or pseudo-Euclidean spaces. Rolling motions provide interesting examples of nonholonomic systems and symmetric spaces appear associated to important applications. We make a connection between the structure of the kinematic equations of rolling and the natural decomposition of the Lie algebra associated to the symmetric space. This emphasises the relevance of Lie theory in the geometry of rolling manifolds and explains why many particular examples scattered through the existing literature always show a common pattern.

    Mathematics Subject Classification: Primary: 70G45, 53C35; Secondary: 53A35, 53A17.


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  • Figure 1.  A sphere $ \mathbf{M} $ is rolling upon a surface $ \mathbf{M}_0 $, along the development curve $ \sigma_0 $, without slipping or twisting

    Figure 2.  A vector attached to a rolling manifold generates a vector field along the path $ \chi(t)(p) $ in the ambient space; thus a tangent (normal) vector in $ \mathbf{T}_p \mathbf{M} $ is isometrically carried to a tangent (normal) vector in $ \mathbf{T}_q(\chi(t_0)( \mathbf{M})) = \mathbf{T}_q \mathbf{M}_0 $

    Figure 3.  A sphere $ {\bf S}^{2} $ is rolling upon $ \mathbf{M}_0 $ along the development curve $ \sigma_0 $ without slipping or twisting; the infinitesimal action $ ({\dot\chi\,\chi {^{-1}}})_* $ is orthogonal, with respect to an $ {\rm Ad}(H) $-invariant inner product onk__ge $ \mathfrak{so}({3}) $, to the Lie algebra of the isotropy group

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