Universality theorems for linkages in homogeneous surfaces

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.