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
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]).
Of course, when \(\mathcal {M}\) is a Riemannian manifold, we may choose all the lengths in \(\mathbb {R}_{\ge 0}\)!
Already defined informally at the beginning of Sect. 1.3.
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 \(\mathcal M^{W \cup P}\).
References
Akbulut, S., King, H.C.: The topology of real algebraic sets with isolated singularities. Ann. Math. (2) 113(3), 425–446 (1981)
Akbulut, S., King, H.: Topology of Real Algebraic Sets, Mathematical Sciences Research Institute Publications, vol. 25. Springer, New York (1992)
Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, Berlin (1998)
Farber, M.: Invitation to topological robotics. European Mathematical Society (EMS), Zurich Lectures in Advanced Mathematics, Zürich (2008)
Jordan, D., Steiner, M.: Configuration spaces of mechanical linkages. Discrete Comput. Geom. 22(2), 297–315 (1999)
Jordan, D., Steiner, M.: Compact surfaces as configuration spaces of mechanical linkages. Israel J. Math. 122(1), 175–187 (2001)
Kempe, A.B.: On a general method of describing plane curves of the nth degree by linkwork. Proc. Lond. Math. Soc. S1–7(1), 213–216 (1876)
King, H.C.: Configuration spaces of linkages in \({\mathbb{R}}^n\). arXiv preprint arXiv:math/9811138 (1998)
Kapovich, M., Millson, J.J.: The relative deformation theory of representations and flat connections and deformations of linkages in constant curvature spaces. Compos. Math. 103(3), 287–317 (1996)
Kapovich, M., Millson, J.J., et al.: On the moduli space of a spherical polygonal linkage. Can. Math. Bull. 42(3), 307–320 (1999)
Kapovich, M., Millson, J.J.: Universality theorems for configuration spaces of planar linkages. Topology 41(6), 1051–1107 (2002)
Mnëv, N.E.: The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In: Topology and Geometry—Rohlin Seminar, volume 1346 of Lecture Notes in Mathematic, pp. 527–543, Berlin, Springer (1988)
Mumford, D.: Algebraic geometry. I. Classics in Mathematics. Springer, Berlin (1995). Complex projective varieties, Reprint of the 1976 edition
Nash, J.: Real algebraic manifolds. Ann. Math. 2(56), 405–421 (1952)
Tognoli, A.: Su una congettura di Nash. Ann. Scuola Norm. Sup. Pisa 3(27), 167–185 (1973)
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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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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DOI: https://doi.org/10.1007/s10711-016-0168-y
Keywords
- Linkage
- Configuration space
- Universality theorem
- Semi-algebraic set
- Algebraic set
- Minkowski plane
- Hyperbolic geometry
- Spherical geometry
- Metric geometry
- Riemannian geometry