Tag Archives: sub-riemannian geometry

The origin of emergent algebras (part II)

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 T_{p}M at a point p \in M, where M 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

[T_{p} M, d^{p}, p] = \lim_{\varepsilon \rightarrow 0} [\bar{B}(p, \varepsilon), \frac{1}{\varepsilon} d, p]

where [X, d, p] is the isometry class of the compact  metric space (X,d) with a marked point p \in X. (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 p \in M. 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 M \times M, Bellaiche proposes to recover the Carnot group algebraic structure of T_{p}M.

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 T_{p}M for regular p” 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 \varepsilon > 0 (but sufficiently small), he uses a function denoted by \phi_{x}, which he states that it is defined on T_{x} M with values in M. 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 T_{x} M, 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 \phi_{x}, 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 \phi_{x} are also decorated with the scale  parameter \varepsilon >0, so they should deserve the better notation \phi_{\varepsilon}^{x},  and that these functions behave like dilations.

A natural alternative to Bellaiche’s proposal would be to use an embodiment of the isometry class [\bar{B}(x, \varepsilon), \frac{1}{\varepsilon} d, x] on the space M, instead of the differential geometric tangent space T_{x}M.  With this choice, what Bellaiche is saying is that we should consider dilation like functions \delta^{x}_{\varepsilon} defined locally from M to M such that:

  • they preserve the point x (which will become the “0” of the metric tangent space): \delta^{x}_{\varepsilon} x = x
  • they form a one-parameter group with respect to the scale: \delta^{x}_{\varepsilon} \delta^{x}_{\mu} y = \delta^{x}_{\varepsilon \mu} y and \delta^{x}_{1} y = y,
  • for any y, z at a finite distance from x (measured with the sub-riemannian distance d, more specifically such that  d(x,y), d(x,z) \leq 1) we have

d^{x}(y,z) = \frac{1}{\varepsilon} d( \delta^{x}_{\varepsilon} y, \delta^{x}_{\varepsilon}z) + O(\varepsilon)

where O (\varepsilon) is uniform w.r.t. (does not depend on) x, y , z in compact sets.

Moreover, we have to keep in mind that the “dilation”  \delta^{x}_{\varepsilon} is defined only locally, so we have to avoid to go far from x, for example we have to avoid to apply the dilation for \varepsilon very big to points at finite distance from x.

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 \varepsilon \rightarrow 0 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 x \in M and two near points y and z, which are fixed. The problem is how to construct an intrinsic addition of y and z with respect to x. Let us denote by y +_{x} z the result we are seeking. (The link with the trivial pair groupoid is that we want to define an operation which takes (x,y) and (x,z) as input and spills (x, y+_{x} z) as output.)

The relevant figure is the following one, which is an improved version of the Figure 5, page 76 of Bellaiche paper.

bellaiche

Bellaiche’s recipe has to do with the points in blue. He says that first we have to go far from x, by dilating the point z w.r.t. the point x, with the coefficient \varepsilon^{-1}. Here \varepsilon is considered to be small (it will go to 0), therefore \varepsilon^{-1} is big.  The result is the blue point A. Then, we dilate (or rather contract) the point A  by the coefficient \varepsilon w.r.t. the point y. The result is the blue point B.

Bellaiche claims that when \varepsilon goes to 0 the point B converges to the sum y +_{x} z. 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 A which is far from x. 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 C, D, E. Indeed, instead of going far away first (the blue point A), then coming back at finite distance from x (the blue point B), we may first come close to x (by using  the red points C, D), then inflate the point D to finite distance from x and get the point E. 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 E = B, 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:

\delta^{x}_{\varepsilon} y = x \circ_{\varepsilon} y

Then, it is easy to verify that the equality between the red point E and the blue point B 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.

The origin of emergent algebras

In the last section “Why is the tangent space a group?” (section 8) of the great article by A. Bellaiche, The tangent space in sub-riemannian geometry*, the author explains a very interesting story, where names of Gromov and Connes appear, which is the first place, to my knowledge, where the idea of emergent algebras appear.

In a future post I shall comment more consistently on the math, but this time let me give you the relevant passages.

[p. 73] “Why is the tangent space a group at regular points? […] I have been puzzled by this question. Drawing a Lie algebra from the bracket structure of some X_{i}‘s did not seem to me the appropriate answer. I remember having, at last, asked M. Gromov about it (1982). The answer came under the form of a little apologue:

Take a map f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}. Define its differential as

(79)              D_{x} f(u) = \lim_{\varepsilon \rightarrow 0} \varepsilon^{-1} \left[ f(x+\varepsilon u) - f(x) \right],

provided convergence holds. Then D_{x}f is certainly homogeneous:

D_{x}f(\lambda u) = \lambda D_{x}f(u),

but it need not satisfy the additivity condition

D_{x}f(u+v) = D_{x}f(u) + D_{x}f(v).

[…] However, if the convergence in (79)  is uniform on some neghbourhood of (x,0)  […]  then D_{x}f is additive, hence linear. So, uniformity was the key. The tangent space at p is a limit, in the [Gromov-]Hausdorff sense, of pointed spaces […] It certainly is a homogeneous space — in the sense of a metric space having a 1-parameter group of dilations. But when the convergence is uniform with respect to p, which is the case near regular points, in addition, it is a group.

Before giving the proof, I want to tell of another, later, hint, coming from the work of A. Connes. He has made significant use of the following observation: The tangent bundle TM to a differentiable manifold M is, like M \times M, a groupoid. […] In fact TM is simply a union of groups. In [8], II.5, it is stated that its structure may be derived from that of M \times M by blowing up the diagonal in M \times M. This suggests that, putting metrics back into the picture, one should have

(83)          TM = \lim_{\varepsilon \rightarrow 0} \varepsilon^{-1} (M \times M)

[…] in some sense to be made precise.

There is still one question. Since the differentiable structure of our manifold is the same as in Connes’ picture, why do we not get the same abelian group structure? One can answer: The differentiable structure is strongly connected to (the equivalence class of) Riemannian metrics; differentiable maps are locally Lipschitz, and Lipschitz maps are almost everywhere differentiable. There is no such connection between differentiable maps and the metric when it is sub-riemannian. Put in another way, differentiable maps have good local commutation properties with ordinary dilations, but not with sub-riemannian dilations \delta_{\lambda}.

So, one should not be abused by (83) and think that the algebraic structure of T_{p}M stems from the absolutely trivial structure of M \times M! It is concealed in dilations, as we shall now prove.

*) in the book Sub-riemannian geometry, eds. A. Bellaiche, J.-J. Risler, Progress in Mathematics 144, Birkhauser 1996

Ado’s theorem for groups with dilations?

Ado’s theorem  is equivalent with the following:

Theorem. Let G be a local Lie group. Then there is a real, finite dimensional vector space V and an injective, local group morphism from (a neighbourhood of the neutral element of) G to GL(V), the linear group of $V$.

Any proof I am aware of, (see this post for one proof and relevant links),  mixes the following ingredients:

–  the Lie bracket and the BCH formula,

– either reduction to the nilpotent case or (nonexclusive) use of differential equations,

– the universal enveloping algebra.

WARNING: further I shall not mention the “local” word, in the realm of spaces with dilations everything is local.

We may pass to the following larger frame of spaces with dilations, dilatation structures or emergent algebras:

– locally compact groups with dilations instead of Lie groups

– locally compact conical groups instead of vector spaces

– linearity in the sense of dilation structures instead of usual linearity.

Conjecture:  For any locally compact group with dilations G there is a locally compact conical group N and an injective morphism \rho: G \rightarrow GL(N) such that for every x \in N the map g \in G \mapsto \rho(g)x is differentiable.

In this frame:

– we don’t have the corresponding Lie bracket and BCH formula, see the related problem of the noncommutative BCH formula,

– what nilpotent means is no longer clear (or needed?)

– we don’t have a clear tensor product, therefore we don’t have a correspondent of the universal enveloping algebra.

Nevertheless I think the conjecture is true and actually much easier to prove than Ado’s theorem, because of the weakening of the conclusion.

Preview of two papers, thanks for comments

Here are two papers:

Local and global moves on planary trivalent graphs, lambda calculus and lambda-Scale (update 03.07.2012, final version, appears as arXiv:1207.0332)

Sub-riemannian geometry from intrinsic viewpoint    (update 14.06.2012: final version, appears as arxiv:1206.3093)

which are still subject to change.  Nevertheless most of what I am trying to communicate is there. I would appreciate  mathematical comments.

This is an experiment,  to see what happens if I make previews of papers available, like a kind of a blog of papers in the making.

Intrinsic characterizations of riemannian and sub-riemannian spaces (I)

In this post I explain what is the problem of intrinsic characterization of riemannian manifolds, in what sense has been solved in full generality by Nikolaev, then I shall comment on the proof of the Hilbert’s fifth problem by Tao.

In the next post there will be then some comments about Gromov’s problem of giving an intrinsic characterization of sub-riemannian manifolds, in what sense I solved this problem by adding a bit of algebra to it. Finally, I shall return to the characterization of riemannian manifolds, seen as particular sub-riemannian manifolds, and comment on the differences between this characterization and Nikolaev’ one.

1. History of the problem for riemannian manifolds. The problem of giving an intrinsic characterization of riemannian manifolds is a classic and fertile one.

Problem: give a metric description of a Riemannian manifold.

Background: A complete riemannian manifold is a length metric space (or geodesic, or intrinsic metric space) by Hopf-Rinow theorem. The problem asks for the recovery of the manifold structure from the distance function (associated to the length functional).

For 2-dim riemannian manifolds the problem has been solved by A. Wald [Begrundung einer koordinatenlosen Differentialgeometrie der Flachen, Erg. Math. Colloq. 7 (1936), 24-46] (“Begrundung” with umlaut u, “Flachen” with umlaut a, sorry for this).

In 1948 A.D. Alexandrov [Intrinsic geometry of convex surfaces, various editions] introduces its famous curvature (which uses comparison triangles)  and proves that, under mild smoothness conditions  on this curvature, one is capable to recover the differential structure and the metric of the 2-dim riemannian manifold. In 1982 Alexandrov proposes as a conjecture that a characterization of a riemannian manifold (of any dimension) is possible in terms of metric (sectional) curvatures (of the type introduced by Alexandrov) and weak smoothness assumptions formulated in metric way (as for example Holder smoothness). Many other results deserve to be mentioned (by Reshetnyak, for example).

2. Solution of the problem by Nikolaev. In 1998 I.G. Nikolaev [A metric characterization of riemannian spaces, Siberian Adv. Math. , 9 (1999), 1-58] solves the general problem of intrinsic characterization of C^{m,\alpha} riemannian spaces:

every locally compact length metric space M, not linear at one of its points,  with \alpha Holder continuous metric sectional curvature of the “generalized tangent bundle” T^{m}(M) (for some $m=1,2,…$, which admits local geodesic extendability, is isometric to a C^{m+2} smooth riemannian manifold..

Therefore:

  • he defines a generalized tangent bundle in metric sense
  • he defines a notion of sectional curvature
  • he asks some metric smoothness of this curvature

and he gets the result.

3. Gleason metrics and Hilbert’s fifth problem. Let us compare this with the formulation of the solution of the Hilbert’s fifth problem by Terence Tao. THe problem is somehow similar, namely recover the differential structure of a Lie group from its algebraic structure. This time the “intrinsic” object is the group operation, not the distance, as previously.

Tao shows that the proof of the solution may be formulated in metric terms. Namely, he introduces a Gleason metric (definition 4 in the linked post), which will turn to be a left invariant riemannian metric on the (topological) group. I shall not insist on this, instead read the post of Tao and also, for the riemannian metric description, read this previous post by me.

Three problems and a disclaimer

In this post I want to summarize the list of problems I am currently thinking about. This is not a list of regular mathematical problems, see the disclaimer on style written at the end of the post.

Here is the list:

1. what is “computing with space“? There is something happening in the brain (of a human or of a fly) which is akin to a computation, but is not a logical computation: vision. I call this “computing with space”. In the head there are a bunch of neurons chirping one to another, that’s all. There is no euclidean geometry, there are no a priori coordinates (or other extensive properties), there are no problems to solve for them neurons, there is  no homunculus and no outer space, only a dynamical network of gates (neurons and their connections). I think that a part of an answer is the idea of emergent algebras (albeit there should be something more than this).  Mathematically, a closely related problem is this: Alice is exploring a unknown space and then sends to Bob enough information so that Bob could “simulate” the space in the lab. See this, or this, or this.

Application: give the smallest hint of a purely relational  model of vision  without using any a priori knowledge of the (euclidean or other) geometry of outer space or any  pre-defined charting of the visual system (don’t give names to neurons, don’t give them “tasks”, they are not engineers).

2. non-commutative Baker-Campbell-Hausdorff formula. From the solution of the Hilbert’s fifth problem we know that any locally compact topological group without small subgroups can be endowed with the structure of a “infinitesimally commutative” normed group with dilations. This is true because  one parameter sub-groups  and Gleason metrics are used to solve the problem.  The BCH formula solves then another problem: from the infinitesimal structure of a (Lie) group (that is the vector space structure of the tangent space at the identity and the maniflod structure of the Lie group) and from supplementary infinitesimal data (that is the Lie bracket), construct the group operation.

The problem of the non-commutative BCH is the following: suppose you are in a normed group with dilations. Then construct the group operation from the infinitesimal data (the conical group structure of the tangent space at identity and the dilation structure) and supplementary data (the halfbracket).

The classical BCH formula corresponds to the choice of the dilation structure coming from the manifold structure of the Lie group.

In the case of a Carnot group (or a conical group), the non-commutative BCH formula should be trivial (i.e. x y = x \cdot y, the equivalent of xy = x+y in the case of a commutative Lie group, where by convention we neglect all “exp” and “log” in formulae).

3. give a notion of curvature which is meaningful for sub-riemannian spaces. I propose the pair curvdimension- curvature of a metric profile. There is a connection with problem 1: there is a link between the curvature of the metric profile and the “emergent Reidemeister 3 move” explained in section 6 of the computing with space paper. Indeed, at page 36 there is this figure. Yes, R^{x}_{\epsilon \mu \lambda} (u,v) w is a curvature!

Disclaimer on style. I am not a problem solver, in the sense that I don’t usually like to find the solution of an already formulated problem. Instead, what I do like to do is to understand some phenomenon and prove something about it in the simplest way possible.  When thinking about a subject, I like to polish the partial understanding I have by renouncing to use any “impure” tools, that is any (mathematical) fact which is strange to the subject. I know that this is not the usual way of doing the job, but sometimes less is more.

Curvdimension and curvature of a metric profile III

I continue from the previous post “Curvdimension and curvature of a metric profile II“.

Let’s see what is happening for (X,g), a sufficiently smooth (C^{4} for example),  complete, connected  riemannian manifold.  The letter “g” denotes the metric (scalar product on the tangent space) and the letter “d” will denote the riemannian distance, that is for any two points x,y \in X the distance  d(x,y) between them is the infimum of the length of absolutely continuous curves which start from x and end in y. The length of curves is computed with the help of the metric g.

Notations.   In this example X is a differential manifold, therefore it has tangent spaces at every point, in the differential geometric sense. Further on, when I write “tangent space” it will mean tangent space in this sense. Otherwise I shall write “metric tangent space” for the metric notion of tangent space.

Let u,v be vectors in the tangent space at $x \in X$. When the basepoint x is fixed by the context then I may renounce to mention it in the various notations. For example \|u\| means the norm of the vector u with respect to the scalar product  g_{x} on the tangent space T_{x} X  at the point x. Likewise,\langle u,v \rangle may be used instead of g_{x}(u,v);  the riemannian curvature tensor at x  may be denoted by R and not by R_{x}, and so on …

Remark 2. The smoothness of the riemannian manifold (X,g) should be just enough such that the curvature tensor is C^{1} and such that for any compact subset C \subset X of X, possibly by rescaling g, the geodesic exponential exp_{x} u makes sense (exists and it is uniquely defined) for any x \in C and for any  u \in T_{x} X with \|u\| \leq 2.

Let us fix such a compact set C and let’s take a  point x \in C.

Definition 5. For any \varepsilon \in (0,1) we define on the closed ball of radius 1 centered at x (with respect to the distance d) the following distance: for any u,v \in T_{x} X with \|u\| \leq 1, \| v\| \leq 1

d^{x}_{\varepsilon} (exp_{x} \, u, exp_{x} v) \, = \, \frac{1}{\varepsilon} d((exp_{x} \, \varepsilon u, exp_{x} \varepsilon v).

(The notation used here is in line with the one used in  dilation structures.)

Recall that the sectional curvature K_{x}(u,v) is defined for any pair of vectors   u,v \in T_{x} X which are linearly independent (i.e. non collinear).

Proposition 1. Let M > 0 be greater or equal than \mid K_{x}(u,v)\mid , for any x \in C and any non-collinear pair of vectors u,v \in T_{x} X with \|u\| \leq 1, \| v\| \leq 1.  Then for any  \varepsilon \in (0,1) and any x \in Cu,v \in T_{x} X with \|u\| \leq 1, \| v\| \leq 1 we have

\mid d^{x}_{\varepsilon} (exp_{x} \, u, exp_{x} v) - \|u-v\|_{x} \mid \leq \frac{1}{3} M \varepsilon^{2} \|u-v\|_{x} \|u\|_{x} \|v\|_{x} + \varepsilon^{2} \|u-v\|_{x} O(\varepsilon).

Corollary 1. For any x \in X the metric space (X,d) has a metric tangent space at x, which is the isometry class of the unit ball in T_{x}X with the distance d^{x}_{0}(u,v) = \|u - v\|_{x}.

Corollary 2. If the sectional curvature at x \in X is non trivial then the metric profile at x has curvdimension 2 and moreover

d_{GH}(P^{m}(\varepsilon, [X,d,x]), P^{m}(0, [X,d,x]) \leq \frac{2}{3} M \varepsilon^{2} + \varepsilon^{2} O(\varepsilon).

Proofs next time, but you may figure them out by looking at the section 1.3 of these notes on comparison geometry , available from the page of  Vitali Kapovitch.