Infinite Bar-Joint Frameworks, Crystals and Operator Theory

We first define continuous flexes and continuous rigidity. The latter means, roughly speaking, that the framework admits no proper deformations that preserve the edge lengths. The definition below gives straightforward generalisations of terms used for finite frameworks. In that case we note that a continuous flex is often referred to as a finite flex while in engineering models it is referred to as a finite mechanism.

Unless we specify otherwise we shall assume that the frameworks under consideration are proper in that the framework points do not lie on a hyperplane in the ambient space ${\mathbb{R}}^{d}$ and that the framework edges $[p_{i},p_{j}]$ have nonzero lengths $|p_{i}-p_{j}|$.

Similarly one defines base-fixed continuous flexes and continuous rigidity for proper frameworks in ${\mathbb{R}}^{d}$ by replacing a base edge by an appropriate set of framework points with maximal affine span.

The simplest kind of continuously rigid framework in the plane is one which is a union of continuously rigid finite frameworks. In particular the following theorem follows simply from the theorem of Laman indicated in the introduction.

By generic, or, more precisely, algebraically generic, we mean, as is usual, that the coordinates of any finite set of framework points is algebraically independent over the rational numbers. Unlike the case of finite frameworks it is possible to construct two generic frameworks with the same abstract graph one of which is flexible and one of which is rigid. Accordingly it seems appropriate to formulate the following definition to extend the usual usage.

Note that if $G$ is rigid and if $H$ is a containing graph for $G$ with the same vertex set then every generic framework $(H,p)$ in the plane is rigid.

It seems likely that the converse to the theorem above holds. That is, if $H$ does not contain a sequentially Laman graph (in the sense below) with the same vertex set, then $H$ is not generically rigid. We comment more on this later in Section 2.6.

Rigidity and flexibility are properties of the entire framework and it is such entire features and their inter-relationships that are of primary interest in what follows. One would like to understand the relationship with small scale or local structure, such as local counting conditions and local connectivity. Additionally, as above, one would like to relate entire properties to sequential features that pertain to an exhausting chain of finite subframeworks and for this the following definition is helpful.

For example, we may refer to an infinite graph as being $\sigma$-Laman, or $\sigma$-Laman$-1$ and an infinite framework as being $\sigma$-rigid. To say that an infinite framework is $\sigma$-flexible, or sequentially flexible, is rather vacuous since it usually prevails for trivial reasons. (One can construct countably infinite edge complete frameworks that may fail to be so but all our examples are sequentially flexible.) The more interesting property is the failure of sequentially rigidity as in the following definition.

An important topic in finite rigidity frameworks is that of global rigidity, also called unique rigidity. This holds, in the two dimensional setting, if there is, up to congruency, only one embedding of the framework in ${\mathbb{R}}^{2}$. This implies, for example, that these frameworks admit no foldings, and indeed have unique diagrams up to rotations and reflection. One can extend the term to infinite frameworks but we do not consider this issue here at all. One might be tempted to say that a rigid infinite framework, especially one with vanishing flexibility, is globally rigid, but we refrain from doing so because of conflict with this usage.

The term ”global” for global rigidity is natural since rigidity for finite frameworks is equivalent to the ”local” property that ”nearby equivalent frameworks are congruent”. That is, if there exists $\epsilon>0$ such that if $(G,p^{\prime})$ is a finite framework with $|p_{i}-p_{i}^{\prime}|<\epsilon$, for all $i$, and if $(G,p^{\prime})$ is equivalent to $(G,p)$, in the sense that corresponding edges have the same length, then $(G,p^{\prime})$ and $(G,p)$ are congruent. See, for example, Gluck [12] and Asimow and Roth [1].

An infinite simple graph $G$ is locally finite if for every vertex $v$ there are finitely many incident edges. Amongst such graphs are those for which there is an upper bound to the degree of the vertices, as in the case of the graphs of crystal frameworks. Within this class a graph $G$ is said to be $r$-regular if every vertex has degree $r$. We remark that the theory of tilings provides a wealth of examples of planar frameworks which are $4$-regular.

The removal of a framework edge from a rigid framework may result in flexibility which is, roughly speaking, of a one-dimensional nature. We reserve the term linkage for such a mathematical object, which we formally specify in the next definition. We remark that finite frameworks are also referred to as linkages, particularly when they are flexible, perhaps with several degrees of freedom, but this should not cause confusion.

A two-sided continuous flex $p(t)$ of $(G,p)$ is defined as above but for the replacement of $[0,1]$ by $[-1,1]$. The following formal definition of an infinite linkage reflects the fact that the initial motion of a base-fixed linkage is uniquely determined by the angle change at any flexible joint.

Many interesting finite linkages were considered in the nineteenth century in connection with mechanical linkages. See, for example, Kempe [20]. Note however that the definition is liberal in allowing coincident joints and self-intersecting flexes. Also the definition refers to local deformation behaviour and this does not rule out the possibility of bifurcations occurring in a parameter extension of the given flex.

It is a simple matter to construct diverse infinite linkages by tower constructions or progressive assembly. (See, for example, the Cantor tree frameworks of [25].) However, some such constructions lead to frameworks with vanishing flexibility and so are not linkages in this case. An elementary illustration is given in Figure 4 wherein a two-way infinite rectangular strip linkage is augmented by adding flex-restricting cross braces in an alternating fashion. If the brace lengths tend to the diagonal length from above then the infinite framework is rigid. Evidently in this case the triangle inequality is playing a role in isolating one real solution to the solution set $V(G,p)$ defined below. One can also construct examples in which this isolation is less evident, with all joint angles bounded away from zero and $\pi/2$ for example.

A more interesting and subtle form of vanishing flexibility is due to progressive flex amplification rather than local flex restrictions. Roughly speaking, if a small flex is initiated at a particular joint and the flex propagates in some amplifying manner, then the triangle inequality at some far remove may prohibit any further increase. If the framework is infinite then no local joint flex may be possible at all. The strip framework of concatenated levers in Figure 5 gives an example where the amplification is evident, while the rigidity of the strip framework in Figure 6 and the trapezium strip in Figure 7 is less evident. The lever framework has a natural infintesimal flex, in the sense of Section 3, which is unbounded. The corresponding flexes of Figures 6 and 7 however, are bounded with amplification unfolding as a second order effect. This is proven in the next subsection.

It is a straightforward matter to incorporate the vanishing flexibility of the strips above as subframeworks of a ${\mathbb{Z}}^{2}$-periodic framework. This process is indicated in Figures 9, 10 and 11 below, where the infinite frameworks are determined as the periodic extensions of the given unit cell. Figure 9 shows a linkage formed as a ”fence lattice” composed of infinite horizontal and vertical $\sigma$-rigid bands. Figure 10 shows an analogue where the infinite bands have been replaced by rigid strip frameworks. Figure 11 is an elaboration of this in which cross braces have been introduced to remove the flexibility. (Only one edge is needed for this whereas the example given is periodic.) The additional degree 2 vertex in the cell ensures that the framework is not $\sigma$-rigid, while the infinite bands remain vanishingly rigid.

Let us note that for the framework in Figure 11, with its curious mixture of rigidity and flexibility, one can add any finite number of additional degree 2 vertices without changing the rigidity of the framework. In particular we have a construction that proves the following proposition.

One can readily extend this fanciful idea in various ways to obtain such structures in higher dimensions. For example, start with a one-dimensionally periodic $\sigma$-rigid girder in 3D and augment it with trapezium ”tents” of alternating height to creates vanishingly rigid girders. Also periodically interpolate any number of degree two vertices into the tent top edges without removing the vanishing flexibility. Join infinitely many such component girders periodically at appropriate tent-top edges to create a fence framework and add linear jointed cross braces to create, finally, a 2D periodic grid which is continuously rigid in 3D, which is not $\sigma$-rigid and which has average coordination number arbitrarily close to two.

For a finite or infinite framework ${\mathcal{G}}=(G,p)$ in ${\mathbb{R}}^{2}$ define the function

This is the usual edge function of the framework and depends only on the abstract graph $G$.

In general the solution set of an infinite framework need not be a real algebraic variety even when it is ”finitely parametrised”. In less wild situations it can be useful to relate $V(G,p)$ to the algebraic variety $V(H,\pi_{H}(p))$ associated with a finite subgraph $H$ of $G$, or with an elementary subgraph such as a tree, or even a set of vertices.

We may also form the following associated notions.

Finite flexibility in the sense above is a strong property in which paths from $p$ in the solution set $V(G,p)$ are determined near $p$ by the finite algebraic variety $V(H,p)$. Note that the term ”infinitely flexible” is not appropriate to describe a flexible framework which is not finitely flexible since it is possible to construct linkages, in our formal sense, which are not finitely flexible. This is the case for the periodic framework in Figure 12 which is a linkage because of partial vanishing flexibility.

An important class of frameworks which appear in mathematical models are those that are distance regular.

For such a framework $(G,p)$ it is natural to consider the nearby frameworks with the same graphs but with slightly perturbed framework points (and therefore edge lengths). If a property holds for all such frameworks, for some perurbation distance $\epsilon$ then we call such a property a stable property for the the framework.

Formally, an $\epsilon$-perturbation of a distance regular framework ${\mathcal{G}}=(G,p)$ is a framework ${\mathcal{G}}^{\prime}=(G,p^{\prime})$ for which $|p_{i}-p_{i}^{\prime}|<\epsilon$, for all corresponding framework points. Recall that a finite framework in ${\mathbb{R}}^{d}$ is said to be $\epsilon$-rigid if it is congruent to every equivalent $\epsilon$-perturbation. Let us say that a general framework is perturbationally rigid if it is $\epsilon$-rigid for some $\epsilon$. It is a well-known fact that perturbational rigidity and rigidity are equivalent in the case of algebraically generic finite frameworks [1], [12]. However, it is straightforward to see that this equivalence thoroughly fails for general infinite frameworks (see [25]).

Likewise, if $P$ is any particular property of a distance-regular infinite framework then we may say that ${\mathcal{G}}$ is stably $P$ if, for some $\epsilon>0$, the property $P$ holds for all $\epsilon$-perturbations.

The argument above also shows that the semi-infinite trapezium strip framework of Figure 7 has a continuous flex but no two-sided flex.

It seems to be a fundamental and interesting issue to determine the ways in which infinite bar-joint frameworks are rigid or continuously flexible. In this section we give some further definitions, we give sufficient conditions for the existence of a proper flex and we contemplate a plausible infinite framework version of Laman’s theorem.

Flexes are often infinitely differentiable or smooth in the sense of the following formal definition. This is the case for example, for the ”alternation” flexes of ${\mathcal{G}}_{sq}$ and ${\mathcal{G}}_{kag}$.

The smoothness of a flex is a local requirement whereas the following terms impose various increasing forms of global constraint. In particular rotational flexes of infinite frameworks with unbounded diameter are not bounded flexes, while a translational flex of an infinite framework is a bounded flex but is not a vanishing flex. Adopting a term that has been used in applications [14] we refer to flexes which are not bounded as colossal flexes.

Also we say that $(G,p)$ has a deformation (resp. bounded or vanishing deformation) if it has a base-fixed flex $p(t)$ (which is bounded or vanishing).

There is a sense in which vanishing flexibility is almost the only obstacle to the existence of a flex of a framework all of whose finite subframeworks are flexible. More precisely, in the hypotheses of the next theorem we assume that there are two distinguished framework vertices, $p_{1},p_{2},$ such that any finite subframework $(H,p)$ containing $p_{1},p_{2}$ has a flex which properly separates this pair in the sense of condition (ii) below. The additional requirement needed is that there is a family of flexes of the finite subframeworks whose restrictions to any given subframework $(H,p)$ are uniformly smooth in the sense of condition (i). Note that the constant here depends only on $(H,p)$ and indeed the resulting flex may of necessity be a colossal flex.

Note that it is essential that the separated vertices of condition (ii) are the same for each subgraph. To see this note that the two-way infinite trapezium strip framework considered in Figure 7 has smooth deformations on each of its finite strip subframeworks, each of which ”separates” some two vertices (at the end of the strip) by a fixed positive distance. Nevertheless the infinite strip fails to have a deformation.

Let us recall a version of Laman’s theorem.

For convenience we refer to a maximally independent finite graph as a Laman graph. We remark that any Laman graph can be obtained from a triangle graph by a sequence of moves known as Henneberg moves. The first of these adds a new vertex with two connecting edges while the second breaks an edge into two at a new vertex which is then connected by a new edge to another point of the graph.

Now let $G$ be an infinite graph which contains a subgraph $H$ on all the vertices of $G$ and suppose that $H$ is $\sigma$-Laman. In view of Laman’s theorem every algebraically generic realisation of $H$ (and hence $G$) in the plane is $\sigma$-rigid and so continuously rigid. Is the converse true ? That is, if every generic realisation of an infinite graph $G$ is rigid does $G$ necessarily contain a $\sigma$-Laman subgraph with the same vertex set.

To see this one needs to show is that if $G$ is $\sigma$-(Laman-1) and not $\sigma$-Laman, then there exists a vertex generic realisation $(G,p)$ in the plane which has a continuous flex. That is, we want to build up a flex of the infinite structure by adding new vertices and edges, in the least handicapping way, to allow all, or most of the flex of an initial finite subgraph to be extended.

Alternatively, and more explicitly, suppose that one starts with a generic connected Laman-1 framework $(G_{1},p)$ with $n$ vertices and an infinite sequence of Henneberg move ”instructions”. These instructions yield a unique infinite graph. Is it possible to choose associated framework points $p_{n},p_{n+1},..$ so judiciously that some (perhaps small) flex of $(G_{1},p)$ extends fully to each successive finite extension framework (and hence the infinite framework) ?

Recall that for a finite framework $(G,p)$ in ${\mathbb{R}}^{d}$ with $n=|V|$ an infinitesimal flex is a vector $u=(u_{1},\dots,u_{n})$ in the vector space ${\mathcal{H}}_{v}={\mathbb{R}}^{d}\oplus\dots\oplus{\mathbb{R}}^{d}$ such that the orthogonality relation $\langle p_{i}-p_{j},u_{i}-u_{j}\rangle=0$ holds for each edge $(v_{i},v_{j})$. This condition ensures that if each $p_{i}$ is perturbed to $p_{i}(t)=p_{i}+tu_{i}$, with $t$ small, then the edge length perturbations are of second order only as $t$ tends to zero. That is, for all edges,

If $q(t):[-1,1]\to{\mathcal{H}}_{v}$ is a two-sided smooth flex of the finite framework $(G,p)$ then $q^{\prime}(0)$ is an infinitesimal flex and for a generic finite framework every infinitesimal flex arises in this way. See Asimow and Roth [1] for example.

Associate with an infinite framework $(G,p)$ the product vector space

consisting of all sequences $u=(u_{1},u_{2},\dots)$. Conceptually such a vector corresponds to a specification of instantaneous velocities, or to a perturbation sequence, applied to the framework joints. Define an infinitesimal flex of $(G,p)$ to be a vector $u$ in ${\mathcal{H}}_{v}$ for which, as above, $\langle p_{i}-p_{j},u_{i}-u_{j}\rangle=0$ holds for each edge $(v_{i},v_{j})$, and let ${\mathcal{H}}_{fl}$ denote the linear space of all these vectors. In the planar case ${\mathcal{H}}_{fl}$ contains the three-dimensional linear subspace (assuming $G$ has at least one edge) of the infinitesimal flexes that arise from the isometries of ${\mathbb{R}}^{2}$. Note that a nonzero rotation infinitesimal flex $u$ is an unbounded sequence if and only if $(G,p)$ is an unbounded framework. We denote the space of rigid body motion infinitesimal flexes as ${\mathcal{H}}_{rig}$.

If $G$ is infinite then ${\mathcal{H}}_{v}$ contains properly the direct sum space ${\mathcal{H}}_{v}^{00}=\sum_{V}\oplus{\mathbb{R}}^{2}$ consisting of vectors whose coordinates are finitely supported, in the sense of being finitely nonzero. The following definition is convenient and evocative.

We now give the usual direct definition of the rigidity matrix $R(G,p)$ of a framework $(G,p)$, allowing $G$ to be infinite. This matrix could also be introduced via the Jacobian of the equation system that defines $V(G,p)$ since $2R(G,p)$ is the Jacobian evaluated at $p$.

Write $p_{i}=(x_{i},y_{i})$, $u_{i}=(u^{x}_{i},u^{y}_{i}),i=1,2,\dots$, and denote the coordinate difference $x_{i}-x_{j}$ by $x_{ij}$. The rigidity matrix is an infinite matrix $R(G,p)$ with rows indexed by edges $e_{1},e_{2},\dots$ and columns labeled by vertices but with multiplicity two, namely $v_{1}^{x},v_{1}^{y},v_{2}^{x},v_{2}^{y},\dots$. Note that any matrix of this shape, with finitely many nonzero entries in each row, provides a linear transformation from ${\mathcal{H}}_{v}$ to ${\mathcal{H}}_{e}=\prod_{E}{\mathbb{R}}$.

In particular a vector $u$ in ${\mathcal{H}}_{v}$ is an infinitesimal flex if and only if $R(G,p)u=0$.

Since it is understood here that $G$ is a locally finite graph a rigidity matrix hase finitely many entries in each column and so its transpose corresponds to a linear transformations from ${\mathcal{H}}_{e}$ to ${\mathcal{H}}_{v}$.

In the finite case a self-stress represents a finite linear dependence between the rows of the rigidity matrix, which one might abbreviate, with language abuse, by saying that the corresponding edges of the framework are linearly dependent. A self-stress vector $w=(w_{e})_{e\in E}$ can be simply related to a vector $b=(b_{e})$ conceived of as a sequence $b=(b_{e})$ of bar tension forces with a resolution, or balance, at each node. Indeed, for such a force vector $b$ the vector $w$ for which $w_{e}=|p_{i}-p_{j}|^{-1}b_{e}$ ($e=(v_{i},v_{j})$) is a stress vector. Thus there is a simple linear relationship between the space of internal stresses and the space of resolving bar tensions. We shall not consider here the more general stress vectors, important in engineering applications, that arise from an external loading vectors.

Let ${\mathcal{H}}_{e}^{00}$ be the space of finitely supported vectors in ${\mathcal{H}}_{e}$. We say that an infinite framework $(G,p)$ is finitely isostatic if it is internally infinitesimally rigid and if the finite support stress space ${\mathcal{H}}_{str}^{00}:={\mathcal{H}}_{str}\cap{\mathcal{H}}_{e}^{00}$ is equal to $\{0\}.$ It is straightforward to see that the grid frameworks ${\mathcal{G}}_{{\mathbb{Z}}^{d}}$, in their ambient spaces, are finitely isostatic, as is the kagome framework.

Between the extremes of infinitesimal rigidity and internal rigidity there are other natural forms of rigidity such as those given in the following definition. Write $\ell^{\infty}$, $\ell^{2}$ and $c_{0}$ to indicate the usual Banach sequence spaces for countable coordinates, and write ${\mathcal{H}}_{e}^{\infty},{\mathcal{H}}_{v}^{\infty},\dots,{\mathcal{H}}_{v}^{0}$ for the corresponding subspaces of ${\mathcal{H}}_{e}$ and ${\mathcal{H}}_{v}$.

There is companion terminology for flexes and stresses. Thus we refer to vectors in ${\mathcal{H}}_{v}^{2}\cap\ker R(G,p)$ as square summable infinitesimal flexes and so on.

Note that we have defined an infinitesimal flex in a local way, being the verbatim counterpart of the usage for finite frameworks. In particular the notion takes no account of the possibility of (second order) amplification or vanishing flexibility.

Our examples above indicate the importance of shift operators and in the next section we see that the bilateral shift operators, in their Fourier transform realisation as multiplication operators, play a central role in the discussion of periodic frameworks.

Let us note here that approximate infinitesimal flexes are natural for infinite frameworks and the operator theoretic perspective.

Let $(G,p)$ be a distance regular framework. Then it is straightforward to show that the rigidity matrix determines a bounded Hilbert space operator $R$. It is the metrical and geometric properties of the action of $R$ and its transpose that have relevance to rigidity theory rather than the spectral theory of $R$. However we do have the almost vacuous statement that the existence of approximate (square-summable) flexes corresponds to the point $0$ belonging to the approximate point spectrum of $R$. For if $0$ lies in the approximate point spectrum then (by definition) $Rv_{n}$ is a null sequence for some sequence of unit vectors $v_{n}$, and approximation of these unit vectors by vectors with finite support yields, after normalisations, an approximate square-summable flex sequence $(u_{n})$.

It is implicit in the matricial function association below that the rigid unit modes of translationally periodic frameworks are tied to the existence of approximate flexes.

We have already observed some properties of the basic examples of the grid framework ${\mathcal{G}}_{{\mathbb{Z}}^{2}}$, the squares framework ${\mathcal{G}}_{sq}$ and the kagome framework, ${\mathcal{G}}_{kag}$. In ${\mathbb{R}}^{3}$ we also have analogues, such as the cube framework ${\mathcal{G}}_{cube}$, the octagon framework ${\mathcal{G}}_{oct}$ and the kagome net framework ${\mathcal{G}}_{knet}$, which consist, respectively, of vertex-joined cubes, octahedra and tetrahedra, with no shared edges or faces, each in a natural periodic arrangement. The frameworks ${\mathcal{G}}_{kag}$, ${\mathcal{G}}_{cube},{\mathcal{G}}_{oct}$ and ${\mathcal{G}}_{knet}$ are polytope body-pin frameworks but we consider the polytope rigid units as bar-joint subframeworks formed by adding some, or perhaps all, internal edges. As such these frameworks are examples of crystal frameworks in the sense of the formal definition below. We first comment on a wider notion of periodicity.

For example, the two-way infinite dyadic cobweb framework of Figure 15 is affinely periodic for the dilation doubling map and the four-fold dihedral group $D_{4}$.

For another example we may take the infinite ${\mathbb{Z}}$-periodic framework in three dimensions for which Figure 16 forms a perspective view down a central axis, with framework vertices $(\pm 1,\pm 1,m),m\in{\mathbb{Z}}$. Here the affine group is an isometry group isomorphic, as a group, to ${\mathbb{Z}}\times C_{2}\times D_{4}$.

To illustrate the following definition observe in Figure 16 a template of six edges and three vertices which generates the kagome framework by the translations associated with the parallelogram unit cell. Borrowing crystallographic terminology we refer to such a template as a motif for the framework and the chosen translation group. The following formal definition gives a convenient way of specifying abstract crystal frameworks.

We denote a crystal framework ${\mathcal{C}}$ by the triple $(F_{v},F_{e},{\mathcal{T}})$ or by the triple $(F_{v},F_{e},{\mathbb{Z}}^{d})$ in the case of integer translations. An associated unit cell for ${\mathcal{C}}$ may be defined as a set which contains $F_{v}$ and no other framework points and for which the translates under the translation isometries are disjoint and partition the ambient space. For example in the case of ${\mathcal{G}}_{{\mathbb{Z}}^{2}}$ we may take the semiopen set $[0,1)^{2}$ or the set $[0,1)\times[1/2,5/2)$ as unit cells. Such parallelepiped unit cells are useful for us for torus models for crystal frameworks. Voronoi cells (Brillouin zones) also play a unit cell role in applications but we shall not need such geometric detail here.

In many applied settings the appropriate framework models have ”short” edges, spanning no more than two adjacent unit cells. Here we allow general edges which may span a chain of adjacent cells.

Recall from elementary crystallography that, modulo orthogonal transformations, there are 14 different forms (or symmetry types) in which a countable set of isolated points can be arranged with translational symmetry throughout three-dimensional space. These arrangements are called the Bravais lattices and the translation group ${\mathcal{T}}$ above corresponds to such a lattice. Thus each point of the framework lies in the Bravais lattice generated by the orbit of its unique corresponding motif vertex under the translational group.

Recall that a general (countable, locally finite, connected) framework ${\mathcal{G}}$ is rigid if there is no base-fixed continuous flex and is boundedly rigid, or boundedly nondeformable, if there is no bounded base-fixed continuous flex $p(t)$. Recall, from Section 2, that bounded flexes are those for which there is an absolute constant $M$ such that for every vertex $v$ the time separation $|p_{v}(t)-p_{v}(0)|$ is bounded by $M$ for all $t$ and all $v$.

We first describe a context for the standard ”alternation” flexes of ${\mathcal{G}}_{sq}$ and ${\mathcal{G}}_{kag}$ and certain periodic flexes of ${\mathcal{G}}_{kag}$ with reduced symmetry. For these nonbounded flexes translational periodicity is maintained but relative to an affine flow of the ambient space. By an affine flow we mean simply a path $t\to A_{t}$ of affine transformations of ${\mathbb{R}}^{d}$ which is pointwise continuous. The simplest such flow in two dimensions is a contracting flow such as $A_{t}(x,y)=((1-tc)x,(1-tc))y$, $0<c<1$. The alternating flexes of ${\mathcal{G}}_{sq}$ and ${\mathcal{G}}_{kag}$ are associated with such a flow.