I continue from the post “The origin of emergent algebras“, which revolves around the last sections of Bellaiche paper The tangent space in sub-riemannian geometry, in the book Sub-riemannian geometry, eds. A. Bellaiche, J.-J. Risler, Progress in Mathematics 144, Birkhauser 1996.
In this post we shall see how Bellaiche proposes to extract the algebraic structure of the metric tangent space at a point , where is a regular sub-riemannian manifold. Remember that the metric tangent space is defined up to arbitrary isometries fixing one point, as the limit in the Gromov-Hausdorff topology over isometry classes of compact pointed metric spaces
where is the isometry class of the compact metric space with a marked point . (Bellaiche’s notation is less precise but his previous explanations clarify that his relations (83), (84) are meaning exactly what I have written above).
A very important point is that moreover, this convergence is uniform with respect to the point . According to Gromov’s hint mentioned by Bellaiche, this is the central point of the matter. By using this and the structure of the trivial pair groupoid , Bellaiche proposes to recover the Carnot group algebraic structure of .
From this point on I shall pass to a personal interpretation of the section 8.2 “A purely metric derivation of the group structure in for regular ” of Bellaiche article. [We don’t have to worry about “regular” points because I already supposed that the manifold is “regular”, although Bellaiche’s results are more general, in the sense that they apply also to sub-riemannian manifolds which are not regular, like the Grushin plane.]
In order to exploit the limit in the sense of Gromov-Hausdorff, he needs first an embodiment of the abstract isometry classes of pointed metric spaces. More precisely, for any (but sufficiently small), he uses a function denoted by , which he states that it is defined on with values in . But doing so would be contradictory with the goal of constructing the tangent space from the structure of the trivial pair groupoid and dilations. For the moment there is no intrinsic meaning of , although there is one from differential geometry, which we are not allowed to use, because it is not intrinsic to the problem. Nevertheless, Bellaiche already has the functions , by way of his lengthy proof (but up to date the best proof) of the existence of adapted coordinates. For a detailed discussion see my article “Dilatation structures in sub-riemannian geometry” arXiv:0708.4298.
Moreover, later he mentions “dilations”, but which ones? The natural dilations he has from the vector space structure of the tangent space in the usual differential geometric sense? This would have no meaning, when compared to his assertion that the structure of a Carnot group of the metric tangent space is concealed in dilations. The correct choice is again to use his adapted coordinate systems and use intrinsic dilations. In fewer words, what Bellaiche probably means is that his functions are also decorated with the scale parameter , so they should deserve the better notation , and that these functions behave like dilations.
A natural alternative to Bellaiche’s proposal would be to use an embodiment of the isometry class on the space , instead of the differential geometric tangent space . With this choice, what Bellaiche is saying is that we should consider dilation like functions defined locally from to such that:
- they preserve the point (which will become the “” of the metric tangent space):
- they form a one-parameter group with respect to the scale: and ,
- for any at a finite distance from (measured with the sub-riemannian distance , more specifically such that ) we have
where is uniform w.r.t. (does not depend on) in compact sets.
Moreover, we have to keep in mind that the “dilation” is defined only locally, so we have to avoid to go far from , for example we have to avoid to apply the dilation for very big to points at finite distance from .
Again, the main thing to keep in mind is the uniformity assumption. The choice of the embodiment provided by “dilations” is not essential, we may take them otherwise as we please, with the condition that at the limit certain combinations of dilations converge uniformly. This idea suggested by Bellaiche reflects the hint by Gromov. In fact this is what is left from the idea of a manifold in the realm of sub-riemannian geometry (because adapted coordinates cannot be used for building manifold structures, due to the fact that “local” and “infinitesimal” are not the same in sub-riemannian geometry, a thing rather easy to misunderstand until you get used to it).
Let me come back to Bellaiche reasoning, in the setting I just explained. His purpose is to construct the operation in the tangent space, i.e. the addition of vectors. Only that the addition has to recover the structure of a Carnot group, as proven by Bellaiche. This means that the addition is not a commutative, but a noncommutative nilpotent operation.
OK, so we have the base point and two near points and , which are fixed. The problem is how to construct an intrinsic addition of and with respect to . Let us denote by the result we are seeking. (The link with the trivial pair groupoid is that we want to define an operation which takes and as input and spills as output.)
The relevant figure is the following one, which is an improved version of the Figure 5, page 76 of Bellaiche paper.
Bellaiche’s recipe has to do with the points in blue. He says that first we have to go far from , by dilating the point w.r.t. the point , with the coefficient . Here is considered to be small (it will go to ), therefore is big. The result is the blue point . Then, we dilate (or rather contract) the point by the coefficient w.r.t. the point . The result is the blue point .
Bellaiche claims that when goes to the point converges to the sum . Also, from this intrinsic definition of addition, all the other properties (Carnot group structure) of the operation may be deduced from the uniformity of this convergence. He does not give a proof of this fact.
The idea of Bellaiche is partially correct (in regards to the emergence of the algebraic properties of the operation from uniformity of the convergence of its definition) and partially wrong (this is not the correct definition of the operation). Let me start with the second part. The definition of the operation has the obvious default that it uses the point which is far from . This is in contradiction with the local character of the definition of the metric tangent space (and in contradiction with the local definition of dilations). But he is wrong from interesting reasons, as we shall see.
Instead, a slightly different path could be followed, figured by the red points . Indeed, instead of going far away first (the blue point ), then coming back at finite distance from (the blue point ), we may first come close to (by using the red points ), then inflate the point to finite distance from and get the point . The recipe is a bit more complicated, it involves three dilations instead of two, but I can prove that it works (and leads to the definition of dilation structures and later to the definition of emergent algebras).
The interesting part is that if we draw, as in the figure here, the constructions in the euclidean plane, then we get , so actually in this case there is no difference between the outcome of these constructions. At further examination this looks like an affine feature, right? But in fact this is true in non-affine situations, for example in the case of intrinsic dilations in Carnot groups, see the examples from the post “Emergent algebra as combinatory logic (part I)“.
Let’s think again about these dilations, which are central to our discussion, as being operations. We may change the notations like this:
Then, it is easy to verify that the equality between the red point and the blue point is a consequence of the fact that in usual vector spaces (as well as in their non-commutative version, which are Carnot groups), the dilations, seen as operations, are self-distributive! That is why Bellaiche is actually right in his definition of the tangent space addition operation, provided that it is used only for self-distributive dilation operations. (But this choice limits the applications of his definition of addition operation only to Carnot groups).
Closing remark: I was sensible to these two last sections of Bellaiche’s paper because I was prepared by one of my previous obsessions, namely how to construct differentiability only from topological data. This was the subject of my first paper, see the story told in the post “Topological substratum of the derivative“, there is still some mystery to it, see arXiv:0911.4619.