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Universality theorems for linkages in homogeneous surfaces

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Abstract

A mechanical linkage is a mechanism made of rigid rods linked together by flexible joints, in which some vertices are fixed and others may move. The partial configuration space of a linkage is the set of all the possible positions of a subset of the vertices. We characterize the possible partial configuration spaces of linkages in the (Lorentz-)Minkowski plane, in the hyperbolic plane and in the sphere. We also give a proof of a differential universality theorem in the Minkowski plane and in the hyperbolic plane: for any compact manifold M, there is a linkage whose configuration space is diffeomorphic to the disjoint union of a finite number of copies of M. In the Minkowski plane, it is also true for any manifold M which is the interior of a compact manifold with boundary.

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Notes

  1. Our definition of semi-algebraic sets is not the standard one, but we know from the Tarski–Seidenberg theorem that the two definitions are equivalent (see [3]).

  2. Of course, when \(\mathcal {M}\) is a Riemannian manifold, we may choose all the lengths in \(\mathbb {R}_{\ge 0}\)!

  3. Already defined informally at the beginning of Sect. 1.3.

  4. We do not require U or \(\pi _W^{-1}(U)\) to be smooth manifolds: recall that a smooth map on \(\pi _W^{-1}(U)\) is, by definition, the restriction of a smooth map defined on the ambient \(\mathcal M^{W \cup P}\).

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Acknowledgments

This paper corresponds to Part I of my Ph.D. thesis: I would like to thank my advisor Abdelghani Zeghib for his help.

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Correspondence to Mickaël Kourganoff.

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Kourganoff, M. Universality theorems for linkages in homogeneous surfaces. Geom Dedicata 185, 35–85 (2016). https://doi.org/10.1007/s10711-016-0168-y

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