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Introduction to the intrinsic rolling with indefinite metric. (English) Zbl 1373.37152

The main topic of the paper is rolling (without slipping or twisting) of one pseudo-Riemannian manifold over another.
The main objects of investigation are two pseudo-Riemannian manifolds \((M,g)\) and \((\widehat M,\widehat g)\) of the same dimension and index. An intrinsic rolling of \(M\) over \(\widehat M\) along a pair of contact curves \(x(t)\in M\) and \(\widehat x(t)\in \widetilde M\) is defined as a curve \(q(t)\) of linear pseudo-isometries between \(T_{x(t)}M\) and \(T_{\widehat x(t)}\widehat M\) such that \(q(t)\) maps the velocity \(\dot x(t)\) to \(\dot{\widehat x}(t)\) (no-slip condition) and that \(g(t)\) relates parallel transports (with respect to the Levi-Civita connections) along \(x(t)\) and \(\widehat x(t)\) (no-twist condition). The main results concern the problem of existence and uniqueness of an intrinsic rolling along a given pair of curves \((x(t),\widehat x(t))\in M\times\widehat M\). This question can be addressed by studying parallel vector fields along \(x(t)\) and \(\widehat x(t)\) (Lemma 1). It turns out that a crucial factor determining the uniqueness of a rolling is the number of independent vector fields simultaneously parallel along a given curve and orthogonal to it (Theorem 1). In particular, if \(x(t)\) and \(\widehat x(t)\) are geodesics, then this number is the dimension of the manifold minus one, and there are no restrictions for the existence of the rolling \(q(t)\) apart from an obvious non-slip condition (Corollary 1). The construction of a rolling can be also formulated as a lifting of a curve \((x(t),\widehat x(t))\) to a curve horizontal with respect to a linear distribution on the bundle of pseudo-isometries \(\text{Iso}(T M,T \widehat M)\) (Proposition 6). This description can be used to study the casual character of the rolling curves (Theorem 3).
If \((M,g)\) and \((\widehat M,\widehat g)\) are embedded in a pseudo-Euclidean space \((\mathbb{R}^m,J_\nu)\) (here \(\nu\) is the index of the metric) one can introduce the notion of extrinsic rolling, which is basically an intrinsic rolling \(q(t)\) plus a curve \(p(t)\) of linear isometries between the normal spaces \(T^\perp_{x(t)} M\) and \(T^\perp_{\widehat x(t)} \widehat M\), relating the parallel transports in the normal bundles (Definition 3). The results discussed above can be easily extended to extrinsic rolling (Theorem 2 and Proposition 5).
I personally found the article interesting and easy to follow. A particular example of a pseudo-sphere \(S^m_1\subset (\mathbb{R}^{m+1},J_1)\) rolling over its affine tangent plane \(x_0+T_{x_0}S^m_1\) was quite helpful in understanding the theory. It also was a good excuse for touching a few related topics from control theory, such as controllability of the rolling motion, geodesic accessibility by rolling, and causality of the rolling curves.

MSC:

37J60 Nonholonomic dynamical systems
53A17 Differential geometric aspects in kinematics
53A35 Non-Euclidean differential geometry
70F25 Nonholonomic systems related to the dynamics of a system of particles
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