Universality theorems for linkages in homogeneous surfaces

TL;DR

This paper characterizes the configuration spaces of mechanical linkages in various geometric surfaces and proves universality theorems showing any manifold can be realized as a linkage configuration space.

Contribution

It provides a comprehensive classification of linkage configuration spaces in Minkowski, hyperbolic, and spherical geometries, and establishes universality theorems in these contexts.

Findings

Configuration spaces characterized for Minkowski, hyperbolic, and spherical surfaces.

Universality theorems proved for manifolds as linkage configuration spaces.

Extension of universality to manifolds with boundary in Minkowski plane.

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.