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 , there is a linkage whose configuration space is diffeomorphic to the disjoint union of a finite number of copies of . In the Minkowski plane, it is also true for any manifold which is the interior of a compact manifold with boundary.
Chapter 1 Introduction and generalities
A mechanical linkage is a mechanism made of rigid rods linked together by flexible joints. Mathematically, we consider a linkage as a marked graph: lengths are assigned to the edges, and some vertices are pinned down while others may move.
A realization of a linkage in a manifold is a mapping which sends each vertex of the graph to a point of , respecting the lengths of the edges. The configuration space is the set of all realizations. Intuitively, the configuration space is the set of all the possible states of the mechanical linkage. This supposes, classically, the ambient manifold to have a Riemannian structure: thus the configuration space may be seen as the space of “isometric immersions” of the metric graph in .
Here we will always deal with (nontrivially) marked connected graphs, that is, a nonempty set of vertices have fixed realizations (in fact, when is homogeneous, considering a linkage without fixed vertices only adds a translation factor to the configuration space). Hence, our configurations spaces will be compact even if is not compact, but rather complete.
1.1. Some historical background
Most existing studies deal with the special case where is the Euclidean plane and some with the higher dimensional Euclidean case (see for instance [Far08] and [Kin98]). There are also studies about polygonal linkages in the standard 2sphere (see [KM99]), or in the hyperbolic plane (see [KM96]).
Universality theorems. When is the Euclidean plane , a configuration space is an algebraic set. This set is smooth for a generic length structure on the underlying graph.
Universality theorems tend to state that, playing with mechanisms, we get any algebraic set of , and any manifold, as a configuration space! In contrast, it is a hard task to understand the topology or geometry of the configuration space of a given mechanism, even for a simple one.
Universality theorems have been announced in the ambient manifold by Thurston in oral lectures, and then proved by Kapovich and Millson in [KM02]. They have been proved in by King [Kin98], and in and in the 2sphere by Mnëv (see [Mnë88] and [KM02]). It is our aim in the present article to prove them in the cases of: the hyperbolic plane , the sphere and the (Lorentz)Minkowski plane . These are simply connected homogeneous pseudoRiemannian surfaces (the list of such spaces includes in addition the Euclidean and the de Sitter planes). Then it becomes natural to ask whether universality theorems hold in a more general class of manifolds, for instance on Riemannian surfaces without a homogeneity hypothesis.
In order to be more precise, it will be useful to introduce partial configuration spaces: for a subset of the vertices of , one defines as the set of realizations of the subgraph induced by that extend to realizations of . One has in particular a restriction map: .
If is a vertex of , its partial configuration space is its workspace, i.e. the set of all its positions in corresponding to realizations of .
Euclidean planar linkages. Now regarding the algebraic side of universality, the history starts (and almost ends) in 1876 with the wellknown Kempe’s theorem [Kem76]:
Theorem 1.1.
Any algebraic curve of the Euclidean plane , intersected with a Euclidean ball, is the workspace of some vertex of some mechanical linkage.
This theorem has the following natural generalization, which we will call the algebraic universality theorem, proved by Kapovich and Millson (see [KM02]):
Theorem 1.2.
Let be a compact semialgebraic subset (see Definition 1.12) of (identified with ). Then, is a partial configuration space of some linkage in . When is algebraic, one can choose such that the restriction map is a smooth finite trivial covering.
When is not a smooth manifold, as usual, by a smooth map on it, we mean the restriction of a smooth map defined on the ambient .
From Theorem 1.2, Kapovich and Millson easily derive the differential universality theorem on the Euclidean plane:
Theorem 1.3.
Any compact connected smooth manifold is diffeomorphic to one connected component of the configuration space of some linkage in the Euclidean plane . More precisely, there is a configuration space whose components are all diffeomorphic to the given differentiable manifold.
Jordan and Steiner also proved a weaker version of this theorem with more elementary techniques (see [JS99]).
How to go from the algebraic universality to the differentiable one? The differentiable universality theorems (Theorems 1.3, 1.5 and 1.7) follow immediately from the algebraic ones (Theorems 1.2, 1.4 and 1.6) once we know which smooth manifolds are diffeomorphic to algebraic sets. In 1952, Nash [Nas52] proved that for any smooth connected compact manifold , one may find an algebraic set which has one component diffeomorphic to . In 1973, Tognoli [Tog73] proved that there is in fact an algebraic set which is diffeomorphic to (a proof may be found in [AK92], or in [BCR98]).
In the noncompact case (in which we will be especially interested), Akbulut and King [AK81] proved that every smooth manifold which is obtained as the interior of a compact manifold (with boundary) is diffeomorphic to an algebraic set. Note that conversely, any (nonsingular) algebraic set is diffeomorphic to the interior of a compact manifold with boundary.
1.2. Results
It is very natural to ask if these algebraic and differential universality theorems can be formulated and proved for configuration spaces in a general target space . Our results suggest this could be true: indeed, we naturally generalize universality theorems to the cases of , and , the Minkowski and hyperbolic planes and the sphere, respectively. Notice that for a general , there is no notion of algebraic subset of ! We will however observe that there is a natural one in the cases we are considering here. In the general case, the question around Kempe’s theorem could be rather formulated as: “Characterize curves in that are workspaces of some vertex of a linkage.”
Minkowski planar linkages. These linkages are studied in Chapter 2. Classically, the structure of needed to define realizations of a linkage is that of a Riemannian manifold. Observe however that a distance, not necessarily of Riemannian type, on would also suffice for this task. But our idea here is instead to relax positiveness of the metric. Instead of a Riemannian metric, we will assume has a pseudoRiemannian one. We will actually restrict ourselves to the simple flat case where is a linear space endowed with a nondegenerate quadratic form, and more specially to the 2dimensional case, that is the Minkowski plane . On the graph side, weights of edges are no longer assumed to be positive numbers. This framework extension is mathematically natural, and may be related to the problem of the embedding of causal sets in physics, but the most important (as well as exciting) fact for us is that configuration spaces are (a priori) no longer compact, and we want to see what new spaces we get in this new setting.
The LorentzMinkowski plane is endowed with a nondegenerate indefinite quadratic form. We denote the “space coordinate” by and the “time coordinate” by .
The configuration space is an algebraic subset (defined by polynomials of degree ) of ( is the number of vertices of ), and similarly a partial configuration space is semialgebraic (see Definiton 1.12). In contrast to the Euclidean case, these sets may be noncompact (even if has some fixed vertices in ). We will prove:
Theorem 1.4.
Let be a semialgebraic subset of (identified with ). Then, is a partial configuration space of some linkage in . When is algebraic, one can choose such that the restriction map is a smooth finite trivial covering.
Somehow, considering Minkowskian linkages is the exact way of realizing noncompact algebraic sets! In particular, Kempe’s theorem extends (globally, i.e. without taking the intersection with balls) to the Minkowski plane: any algebraic curve is the workspace of one vertex of some linkage.
Remark.
If the restriction map is injective, then it is a bijective algebraic morphism from to , but not necessarily an algebraic isomorphism. In fact, it is true for nonsingular complex algebraic sets that bijective morphisms are isomorphisms, but this is no longer true in the real algebraic case (see for instance [Mum95], Chapter 3).
We also have a differential version of the universality theorem in the Minkowski plane (which follows directly from Theorem 1.4, as explained at the end of Section 1.1):
Theorem 1.5.
For any differentiable manifold with finite topology, i.e. diffeomorphic to the interior of a compact manifold with boundary, there is a linkage in the Minkowski plane with a configuration space whose components are all diffeomorphic to . More precisely, there is a partial configuration space which is diffeomorphic to and such that the restriction map is a smooth finite trivial covering.
Hyperbolic planar linkages. In Chapter 3, we prove that both algebraic and differential universality theorems hold in the hyperbolic plane. The problem is that the notion of algebraic set has no intrinsic definition in the hyperbolic plane. However, it is possible to define an algebraic set in the Poincaré halfplane model (and hence in ) as an algebraic set of which is contained in the halfplane. In fact, it turns out that the analogous definitions in the other usual models (the Poincaré disc model, the hyperboloid model, or the BeltramiKlein model) are all equivalent. With this definition, we obtain the same results as in the Euclidean case:
Theorem 1.6.
Let be a compact semialgebraic subset of (identified with a subset of using the Poincaré halfplane model). Then, is a partial configuration space of some linkage in . When is algebraic, one can choose such that the restriction map is a smooth finite trivial covering.
Conversely, any partial configuration space of any linkage with at least one fixed vertex is a compact semialgebraic subset of , so this theorem characterizes the sets which are partial configuration spaces (see Definiton 1.12 for the notion of “semialgebraic”).
In particular, Kempe’s theorem holds in the hyperbolic plane.
And here follows the differential version:
Theorem 1.7.
For any compact differentiable manifold , there is a linkage in the hyperbolic plane with a configuration space whose components are all diffeomorphic to . More precisely, there is a partial configuration space which is diffeomorphic to and such that the restriction map is a smooth finite trivial covering.
Spherical linkages. These linkages are the subject of Chapter 4. In 1988, Mnëv [Mnë88] proved that the algebraic and differential universality theorems hold true in the real projective plane endowed with its usual metric as a quotient of the standard sphere. Even better, he showed that the number of copies in the differential universality for can be reduced to , i.e. any manifold is the configuration space of some linkage. As Kapovich and Millson pointed out [KM02], a direct consequence of Mnëv’s theorem is the differential universality theorem for the 2sphere (but, this time, we get several copies of the desired manifold):
Theorem 1.8 (MnëvKapovichMillson).
For any compact differentiable manifold , there is a linkage in the sphere with a configuration space whose components are all diffeomorphic to .
However, it seems impossible to use Mnëv’s techniques to prove the algebraic universality for spherical linkages: for example, all the configuration spaces of his linkages are symmetric with respect to the origin of . In order to obtain any semialgebraic set as a partial configuration space, we need to start again from scratch and construct linkages specifically for the sphere.
Contrary to the Minkowski and hyperbolic cases, the generalization of the theorems to higher dimensional spheres is straightforward. Thus, we are able to prove the following:
Theorem 1.9.
Let and let be a compact semialgebraic subset of (identified with a subset of ). Then, is a partial configuration space of some linkage in .
In particular, Kempe’s theorem holds in the sphere.
Conversely, any partial configuration space of any linkage is a compact semialgebraic subset of (see Section 1.4), so this theorem characterizes the sets which are partial configuration spaces.
Let us note that even when is algebraic, our construction does not provide a linkage such that the restriction map is a smooth finite trivial covering. We do not know whether such a linkage exists.
Some questions. Our results suggest naturally – among many questions – the following:

Besides the 2dimensional case, are the results in the Minkowski plane true for any (finitedimensional) linear space endowed with a nondegenerate quadratic form? And what about higherdimensional hyperbolic spaces? It is likely that the adaptation of the dimensional proof hides no surprise, like in the Euclidean case, but it would probably require tedious work to prove it.

In all the universality theorems that we prove, we obtain a linkage whose configuration space is diffeomorphic to the sum of a finite number of copies of the given manifold . Is it possible to choose this sum trivial, that is, with exactly one copy of ? (This question is also open in the Euclidean plane.)

Is the differential universality theorem true on any Riemannian manifold?
Linkages on Riemannian manifolds. Let us give a partial answer to the last question using the following idea: just as the surface of the earth looks flat to us, any Riemannian manifold will almost behave as the Euclidean space if one considers a linkage which is small enough. However, our linkage has to be robust to small perturbations of the lengths, which is not the case for many of the linkages described in this paper (consider for example the rigidified square linkage).
Theorem 1.10.
Consider a linkage in the Euclidean space , with at least one fixed vertex, such that for any small perturbation of the length vector , the configuration space remains the same up to diffeomorphism. Then for any Riemannian manifold , there exists a linkage in whose configuration space is diffeomorphic to .
Corollary 1.11.
In any Riemannian surface , the differentiable universality theorem is true for compact orientable surfaces. In other words, any compact orientable surface is diffeomorphic to the configuration space of some linkage .
This leads to the following
Question.
Which manifolds can be obtained as the configuration space of some linkage in which is robust to small perturbations (in the sense of Theorem 1.10) ?
This question is probably very difficult, but it is clear that there are restrictions on such manifolds: for example, they have to be orientable (see again [JS01]).
1.3. Ingredients of the proofs
There are essentially three technical as well as conceptual tools: functional linkages, combination of elementary linkages, and regular inputs. The main idea is always the same as in all the known proofs of Universality theorems (see the proofs of Thurston, Mnëv [Mnë88], King [Kin98] or Kapovich and Millson [KM02]): one combines elementary linkages to construct a “polynomial linkage”.
Functional linkages. One major ingredient in the proofs is the notion of functional linkages. Here we enrich the graph structure by marking two new vertex subsets and playing the role of inputs and outputs, respectively. If the partial realization of is determined by the partial realization of , by means of a function (called the inputoutput function), then we say that we have a functional linkage for (for us, will be the Minkowski plane , the hyperbolic plane or the sphere ). The Peaucellier linkage is a famous historical example: it is functional for an inversion with respect to a circle.
Combination. Another major step in the proofs consists in proving the existence of functional linkages associated to any given polynomial . This will be done by “combining” elementary functional linkages. We define combination so that combining two functional linkages for the functions and provides a functional linkage for .
Elementary linkages. All the work then concentrates in proving the existence of linkages for suitable elementary functions (observe that even for elementary linkages one uses a combination of more elementary ones). As an example, we give the list of the elementary linkages needed to prove Theorem 1.4 (in the Minkowski case):

The linkages for geometric operations:

The robotic arm linkage (Section 2.2.1): one of the most basic linkages, used everywhere in our proofs and in robotics in general.

The rigidified square (Section 2.2.2): a way of getting rid of degenerate configurations of the square using a wellknown construction.

The Peaucellier inversor (Section 2.2.3): this famous linkage of the 1860’s has a slightly different behavior in the Minkowski plane but achieves basically the same goal.

The partial line linkage (Section 2.2.4): it is obtained using a Peaucellier linkage, but does not trace out the whole line.

The integer linkage (Section 2.2.5): it is a linkage with a discrete configuration space.

The line linkage (Section 2.2.6): it draws the whole line, and is obtained by combining the two previous linkages.

The horizontal parallelizer (Section 2.2.7): it forces two vertices to have the same ordinate, and it is obtained by combining several line linkages.

The diagonal parallelizer (Section 2.2.8): its role is similar to the horizontal parallelizer but its construction is totally different.


The linkages for algebraic operations, which realize computations on the line:

The average function linkage (Section 2.3.1): it computes the average of two numbers, and is obtained by combining several of the previous linkages.

The adder (Section 2.3.2): it is functional for addition on the line, and is obtained from several average function linkages.

The square function linkage (Section 2.3.3): it is functional for the square function and is obtained by combining the Peaucellier linkage (which is functional for inversion) with adders. This linkage is somewhat difficult to obtain because we want the inputs to be able to move everywhere in the line, while the inversion is of course not defined at .

The multiplier (Section 2.3.4): it is functional for multiplication and is obtained from square function linkages.

The polynomial linkage (Section 2.3.5): obtained by combining adders and multipliers, it is functional for a given polynomial function . This linkage is used to prove the universality theorems: if the outputs are fixed to , the inputs are allowed to move exactly in .

Regular inputs. In our theorems, we need the restriction map to be a smooth finite trivial covering. In the differential universality Theorem, it ensures in particular that the whole configuration space consists in several copies of the given manifold . The set of regular inputs is the set of all realizations of the inputs which admit a neighborhood onto which the restriction map is a smooth finite covering. We have to be very careful, because even for quite simple linkages such as the robotic arm, the restriction map is not a smooth covering everywhere! There are mainly two possible reasons for the restriction map not to be a smooth covering:

One realization of the inputs may correspond to infinitely many realizations of the whole linkage (for example, when the robotic arm in Section 2.2.1 has two inputs fixed at the same location, the workspace of the third vertex is a whole circle).

Even if it corresponds only to a finite number of realizations, these realizations may not depend smoothly on the inputs (for example, when the robotic arm in Section 2.2.1 is stretched).
New difficulties in each case. While the idea is always the same in all known proofs of universality theorems for linkages, i.e. combine elementary linkages to form a functional linkage for polynomials, each case has its own new difficulties due to different geometric properties, and the elementary linkages always require major changes to work correctly. Here follow examples of such differences with the Euclidean case:
The Minkowski case

The Minkowski plane is not isotropic: its directions are not all equivalent. Indeed, these directions have a causal character in the sense that they may be spacelike, lightlike or timelike. For example, one needs different linkages in order to draw spacelike, timelike and lightlike lines.

In the Euclidean plane, two circles and intersect if and only if , but in the Minkowski plane, the condition of intersection is much more complicated to state (see Section 2.1.2).

In the Euclidean plane, one only has to consider compact algebraic sets. Applying a homothety, one may assume such a set to be inside a small neighborhood of zero, which makes the proof easier. Here, the algebraic sets are no longer compact, so we have to work with mechanisms which are able to deal with the whole plane.
The hyperbolic case

The rigidified square linkage, used extensively in all known proofs in the flat case, does not work anymore in its usual form, and does not have a simple analogue.

There is no natural notion of homothety: in particular, the pantograph does not compute the middle of a hyperbolic segment, contrary to the flat cases.

The notion of algebraic set is less natural than in the flat case.

In every standard model of the hyperbolic plane (such as the Poincaré halfplane), the expression of the distance between two points is much more complicated than in the flat case.
The spherical case

Just as in the hyperbolic case, the curvature prevents the rigidified square linkage from working correctly.

There is no natural notion of homothety.

In the Euclidean or hyperbolic planes, we only need to prove algebraic universality for bounded algebraic sets, which means that our functional linkages do not need to work on the whole surface. In the sphere, all the distances are uniformly bounded (even the lengths of the edges of our linkages), so we need to take into account the whole sphere when constructing linkages.

The compactness of the sphere also makes it difficult to construct linkages which deal with algebraic operations (addition, multiplication, division) since there is no proper embedding of in the sphere.
1.4. Algebraic and semialgebraic sets
In this section, we recall the standard definitions of algebraic and semialgebraic sets. We adapt them to the Minkowski plane, the hyperbolic plane and the sphere in a natural way and state some of their properties.
Definition 1.12.
An algebraic subset of is a set such that there exist and a polynomial such that .
We define a semialgebraic subset of (see [BCR98]) as the projection of an algebraic set^{1}^{1}1Our definition of semialgebraic sets is not the standard one, but we know from the Tarski–Seidenberg theorem that the two definitions are equivalent (see [BCR98]).. More precisely, it is a set such that there exists and an algebraic set of such that , where is the projection onto the first coordinates
We define the (semi)algebraic subsets of by identifying with .
We also define the (semi)algebraic subsets of , using the Poincaré halfplane model (see Definition 3.1), as the (semi)algebraic subsets of which are contained in .
Finally, a (semi)algebraic subset of (for ) is a semialgebraic subset of which is contained in the unit sphere of .
Proposition 1.13.
For any compact semialgebraic subset of , there exists and a compact algebraic subset of such that , where is the projection onto the first coordinates: .
Proof.
First case. Assume for the moment that there exist polynomials such that
Let
and . Then the projection of onto the first coordinates is obviously . Moreover, is compact since it is the image of by the continuous function
General case. The finiteness theorem for semialgebraic sets (see [BCR98], 2.7.2) states that any closed algebraic set can be described as the union of a finite number of sets which satisfy the assumption of the first case: apply the first case to each of the ’s to end the proof. ∎
We end this section with two analogous propositions for the hyperbolic plane and the sphere.
Proposition 1.14.
For any compact semialgebraic subset of , there exists and a compact algebraic subset of (with some ) such that , where is the projection onto the first coordinates: .
Proof.
Let be a compact algebraic set of (with some ) such that , where is the projection onto the first coordinates: . Then the projection of the compact algebraic set
(where ) is exactly . ∎
Proposition 1.15.
For any compact semialgebraic subset of , there exists and a (compact) algebraic subset of (with some ) such that , where is the projection onto the first coordinates: .
Proof.
Let be a compact algebraic set of (with some ) such that , where is the projection onto the first coordinates: . Since is compact, there is a such that , where is the projection onto the last coordinates: . Then the projection of the compact algebraic set
(where ) is exactly . ∎
Of course, Proposition 1.15 extends to with any .
1.5. Generalities on linkages
In the present section, we develop generalities on linkages which apply to the Minkowski plane, the hyperbolic plane and the sphere. Thus, we consider a smooth manifold endowed with a distance function
In the case of a Riemannian manifold (in particular, for the hyperbolic plane and the sphere), the metric determines a realvalued distance on .
In the case of the Minkowski plane, is the plane . Here, we argue by a naive algebraic analogy and define a distance as
Accordingly, the length structure of the linkage will be generalized by taking values in (instead of ) as follows:
Definition 1.16.
A linkage in is a graph together with:

A function (which gives the length of each edge^{2}^{2}2Of course, when is a Riemannian manifold, we may choose all the lengths in !);

A subset of fixed vertices (represented by on the figures);

A function which indicates where the vertices of are fixed;
When the linkage is named , we usually write and name its vertices . If the linkage is a copy of the linkage , the vertex corresponds to the vertex , and so on.
Definition 1.17.
Let be a linkage in . A realization of a linkage in is a function such that:

For each edge , ;

.
Remark.
On the figures of this paper, linkages are represented by abstract graphs. The edges are not necessarily represented by straight segments, and the positions of the vertices on the figures do not necessarily correspond to a realization (unless otherwise stated).
Definition 1.18.
Let be a linkage in . Let . The partial configuration space of in with respect to is
In other words, is the set of all the maps which extend to realizations of . In particular, the configuration space is the set of all realizations of .
Definition 1.19.
A marked linkage is a tuple , where and are subsets of : is called the “input set” and its elements, called the “inputs”, are represented by on the figures, whereas is called the “output set” and its elements, called the “outputs”, are represented by on the figures.
The input map is the map induced by the projection (the restriction map). In other words, for all , we have .
Likewise, we define the output map by .
The notion of marked linkage is not necessary to study configuration spaces. However, in the linkages we use in our proofs, some vertices play an important role (the inputs and the outputs) while others do not: this is why we always consider marked linkages. The following notion^{3}^{3}3already defined informally at the beginning of Section 1.3 accounts for the names “inputs” and “outputs”:
Definition 1.20.
We say that is a functional linkage for the inputoutput function if
1.6. Regularity
Definition 1.21.
Let be a linkage. Let and . Let be the restriction map
We say that is a regular input for if there exists an open neighborhood of such that is a finite smooth covering^{4}^{4}4We do not require or to be smooth manifolds: recall that a smooth map on is, by definition, the restriction of a smooth map defined on the ambient ..
We write the set of regular inputs for . When is the set of all vertices, we simply write .
Roughly speaking, is a regular input for if it determines a finite number of realizations of , and if these configurations are determined smoothly with respect to (in other words, is a smooth multivalued function in a neighborhood of ).
The following fact is simple but essential:
Fact 1.22.
For any , we have
Therefore, in practice, in order to prove that , we only have to prove that for all .
1.7. Changing the input set
In this proposition, we take a linkage, then consider the same linkage with a different set of inputs and analyse the impact on .
Fact 1.23.
Let , and define
Recall that and are the respective input maps of and . Then contains
Proof.
This is a simple consequence of the fact that the composition of two smooth functions is a smooth function. ∎
1.8. Combining linkages
This notion is essential to construct complex linkages from elementary ones. The proofs in this section are straightforward and left to the reader.
Let and be two linkages, , and .
The idea is to construct a new linkage as follows:
Step 1
Consider , the disjoint union of the two graphs and .
Step 2
Identify some vertices of with some vertices of via , without removing any edge.
Since linkages are graphs which come with an additional structure, we need to clarify what happens to the other elements (, , , , ). In particular, note that the inputs of which are in are not considered as inputs in the new linkage .
Definition 1.24 (Combining two linkages).
We define
in the following way:

;

; 
For all , , , define

;

;

;

.
The combination of two linkages is prohibited in the following cases:

There exist such that and (two vertices are fixed at different places but should be attached to the same other vertex).

There exist such that , , and (two edges of different lengths should join one couple of vertices).

There exist and , such that , , , and (again, two edges of different lengths should join one couple of vertices).
Example.
Consider the two identical linkages and :
The inputs of are and the output is .
To combine the two linkages, let and . Then is the following linkage:
The inputs of are and the output is .
We end this section with three facts whose proofs are straightforward. The first describes when is obtained as the combination of two linkages, the second one describes , while the third one establishes a link between the combination of functional linkages and the composition of functions.
Fact 1.25.
Fact 1.26.
Let , be two linkages, , , and . Suppose that satisfies both of the following properties:

;

.
Then .
Fact 1.27.
Let , be two linkages with .
Assume that is a functional linkage for and that is a functional linkage for . Let , a bijection, and . The bijection induces a bijection between and .
Then is functional for .
1.9. Appendix: Linkages on any Riemannian manifold
The aim of this section is to prove Theorem 1.10.
Consider a linkage in the Euclidean space as in the statement of the theorem: we may assume without loss of generality that is a connected graph, that the sum of the lengths of the edges is smaller than , and that one of the vertices is fixed to , so that the configuration space of is a subset of , where is the unit ball of . We introduce the set of all mappings such that (namely, those which map the fixed points to their assigned locations), and define the mapping
Then the configuration space of in is . Making a small perturbation of , we may assume by the Lemma of Sard that is a regular value of . By assumption, this perturbation does not change , up to diffeomorphism.
Let be an open neighborhood of in , equipped with a metric , such that is isometric to an open subset of the Riemannian manifold , and denote by the associated distance on . Applying a linear transformation to , we may assume that (the metric at ) coincides with the canonical Euclidean scalar product on .
For a small enough , the mapping
is welldefined, smooth, and may be extended smoothly to (apply Taylor’s formula).
Then for all small enough ,
where . Notice that is diffeomorphic to the configuration space of some linkage in , since is isometric to an open set of .
The key to the proof is the following fact:
Fact 1.28.
For all , .
Proof.
In this proof, for any open set , we will write the set of paths which take their values in .
Let . For any small enough , we have:
Any path from to takes it values in for some . Thus, taking the limit as , we obtain: