Tag Archives: sub-riemannian geometry

How non-commutative geometry does not work well when applied to non-commutative analysis

I expressed several times the belief that sub-riemannian geometry represents an example of a mathematically new phenomenon, which I call “non-commutative analysis”. Repeatedly happened that apparently general results simply don’t work well when applied to sub-riemannian geometry. This “strange” (not for me) phenomenon leads to negative statements, like rigidity results (Mostow, Margulis), non-rectifiability results (like for example the failure of the theory of metric currents for Carnot groups).  And now, to this adds the following,  arXiv:1404.5494 [math.OA]

“the unexpected result that the theory of spectral triples does not apply to the Carnot manifolds in the way one would expect. [p. 11] ”

i.e.

“We will prove in this thesis that any horizontal Dirac operator on an arbitrary Carnot manifold cannot be hypoelliptic. This is a big difference to the classical case, where any Dirac operator is elliptic. [p. 12]”

It appears that the author reduces the problems to the Heisenberg groups. There is a solution, then, to use

R. Beals, P.C. Greiner, Calculus on Heisenberg manifolds, Princeton University Press, 1988

which gives something resembling spectral triples, but not quite all works, still:

“and show how hypoelliptic Heisenberg pseudodifferential operators furnishing a spectral triple and detecting in addition the Hausdorff dimension of the Heisenberg manifold can be constructed. We will suggest a few concrete operators, but it remains unclear whether one can detect or at least estimate the Carnot-Caratheodory metric from them. [p. 12]”

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This seems to be an excellent article, more than that, because it is a phd dissertation  many things are written clearly.

I am not surprised at all by this, it just means that, as in the case with the metric currents, there is an ingredient in the spectral triples theory which introduces by the backdoor some commutativity, which messes then with the non-commutative analysis  (or calculus).

Instead I am even more convinced than ever that the minimal (!) description of sub-riemannian manifolds, as models of a non-commutative analysis, is given by dilation structures, explained most recently in arXiv:1206.3093 [math.MG].

A corollary of this is: sub-riemannian geometry (i.e. non-commutative analysis of dilation structures)  is more non-commutative than non-commutative geometry .

I’m waiting for a negative result concerning the application of quantum groups to sub-riemannian geometry.

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Sometimes an anonymous review is “a tale told by an idiot …”

… “full of sound and fury, signifying nothing.” And the editor believes it, even if it is self-contradictory, after sitting on the article for half a year.

There are two problems:

  • the problem of time; you write a long and dense article, which may be hard to review and the referee, instead of declining to review it, it keeps it until the editor presses him to write a review, then he writes some fast, crappy report, much below the quality of the work required.
  • the problem of communication: there is no two way communication with the author. After waiting a considerable amount of time, the author has nothing else to do than to re-submit the article to another journal.

Both problems could be easily solved by open peer-review. See Open peer-review as a service.

The referee can well be anonymous, if he wishes, but a dialogue with the author and, more important, with other participants could only improve the quality of the review (and by way of consequence, the quality of the article).

I reproduce further such a review, with comments. It is about the article “Sub-riemannian geometry from intrinsic viewpoint” arXiv:1206.3093 .  You don’t need to read it, maybe excepting the title, abstract and contents pages, which I reproduce here:

Sub-riemannian geometry from intrinsic viewpoint
Marius Buliga
Institute of Mathematics, Romanian Academy
P.O. BOX 1-764, RO 014700
Bucuresti, Romania
Marius.Buliga@imar.ro
This version: 14.06.2012

Abstract

Gromov proposed to extract the (differential) geometric content of a sub-riemannian space exclusively from its Carnot-Caratheodory distance. One of the most striking features of a regular sub-riemannian space is that it has at any point a metric tangent space with the algebraic structure of a Carnot group, hence a homogeneous Lie group. Siebert characterizes homogeneous Lie groups as locally compact groups admitting a contracting and continuous one-parameter group of automorphisms. Siebert result has not a metric character.
In these notes I show that sub-riemannian geometry may be described by about 12 axioms, without using any a priori given differential structure, but using dilation structures instead.
Dilation structures bring forth the other intrinsic ingredient, namely the dilations, thus blending Gromov metric point of view with Siebert algebraic one.
MSC2000: 51K10, 53C17, 53C23

1 Introduction       2
2 Metric spaces, groupoids, norms    4
2.1 Normed groups and normed groupoids      5
2.2 Gromov-Hausdorff distance     7
2.3 Length in metric spaces       8
2.4 Metric profiles. Metric tangent space      10
2.5 Curvdimension and curvature     12

3 Groups with dilations      13
3.1 Conical groups     14
3.2 Carnot groups     14
3.3 Contractible groups   15

4 Dilation structures  16
4.1 Normed groupoids with dilations     16
4.2 Dilation structures, definition    18

5 Examples of dilation structures 20
5.1 Snowflakes, nonstandard dilations in the plane    20
5.2 Normed groups with dilations    21
5.3 Riemannian manifolds    22

6 Length dilation structures 22
7 Properties of dilation structures    24
7.1 Metric profiles associated with dilation structures    24
7.2 The tangent bundle of a dilation structure    26
7.3 Differentiability with respect to a pair of dilation structures    29
7.4 Equivalent dilation structures     30
7.5 Distribution of a dilation structure     31

8 Supplementary properties of dilation structures 32
8.1 The Radon-Nikodym property    32
8.2 Radon-Nikodym property, representation of length, distributions     33
8.3 Tempered dilation structures    34
9 Dilation structures on sub-riemannian manifolds   37
9.1 Sub-riemannian manifolds    37
9.2 Sub-riemannian dilation structures associated to normal frames     38

 

10 Coherent projections: a dilation structure looks down on another   41
10.1 Coherent projections     42
10.2 Length functionals associated to coherent projections    44
10.3 Conditions (A) and (B)     45

11 Distributions in sub-riemannian spaces as coherent projections    45
12 An intrinsic description of sub-riemannian geometry    47
12.1 The generalized Chow condition     47
12.2 The candidate tangent space    50
12.3 Coherent projections induce length dilation structures  53

Now the report:

 

Referee report for the paper


 Sub-riemannian geometry from intrinsic viewpoint

Marius Buliga
for

New York Journal of Mathematics (NYJM).

One of the important theorems in sub-riemannian geometry is a result
credited to Mitchell that says that Gromov-Hausdorff metric tangents
to sub-riemannian manifolds are Carnot groups.
For riemannian manifolds, this result is an exercise, while for
sub-riemannian manifolds it is quite complicate. The only known
strategy is to define special coordinates and using them define some
approximate dilations. With this dilations, the rest of the argument
becomes very easy.
Initially, Buliga isolates the properties required for such dilations
and considers
more general settings (groupoids instead of metric spaces).
However, all the theory is discussed for metric spaces, and the
groupoids leave only confusion to the reader.
His claims are that
1) when this dilations are present, then the tangents are Carnot groups,
[Rmk. The dilations are assumed to satisfy 5 very strong conditions,
e.g., A3 says that the tangent exists – A4 says that the tangent has a
multiplication law.]
2) the only such dilation structures (with other extra assumptios) are
the riemannian manifolds.
He misses to discuss the most important part of the theory:
sub-riemannian manifolds admit such dilations (or, equivalently,
normal frames).
His exposition is not educational and is not a simplification of the
paper by Mitchell (nor of the one by Bellaiche).




The paper is a cut-and-past process from previous papers of the
author. The paper does not seem reorganised at all. It is not
consistent, full of typos, English mistakes and incomplete sentences.
The referee (who is not a spellchecker nor a proofread) thinks that
the author himself could spot plenty of things to fix, just by reading
the paper (below there are some important things that needs to be
fixed).


The paper contains 53 definitions – fifty-three!.
There are 15 Theorems (6 of which are already present in other papers
by the author of by other people. In particular 3 of the theorems are
already present in [4].)
The 27 proofs are not clear, incomplete, or totally obvious.

The author consider thm 8.10 as the main result. However, after
unwrapping the definitions, the statement is: a length space that is
locally bi-lipschitz to a commutative Lie group is locally
bi-lipschitz to a Riemannian manifold. (The proof refers to Cor 8.9,
which I was unable to judge, since it seems that the definition of
“tempered” obviously implies “length” and “locally bi-lipschitz to the
tangent”)


The author confuses the reader with long definitions, which seems very
general, but are only satisfied by sub-riemannian manifolds.
The definitions are so complex that the results are tautologies, after
having understood the assumptions. Indeed, the definitions are as long
as the proofs. Just two examples: thm 7.1 is a consequence of def 4.4,
thm 9.9 is a consequence of def 9.7.

Some objects/notions are not defined or are defined many pages after
they are used.



Small remarks for the author:

def 2.21 is a little o or big O?


page 13 line 2. Which your convention, the curvdim of a come in infinite.
page 13 line -2. an N is missing in the norm


page 16 line 2, what is \nu?

prop 4.2 What do you mean with separable norm?

page 18 there are a couple of “dif” which should be fixed.
in the formula before (15), A should be [0,A]

pag 19 A4. there are uncompleted sentences.

Regarding the line before thm 7.1, I don’t agree that the next theorem
is a generalisation of Mitchell’s, since the core of his thm is the
existence of dilation structures.

Prop 7.2 What is a \Gamma -irq

Prop 8.2 what is a geodesic spray?

Beginning of sec 8.3 This is a which -> This is a

Beginning of sec 9 contains a lot of English mistakes.

Beginning of sec 9.1 “we shall suppose that the dimension of the
distribution is globally constant..” is not needed since the manifold
is connected

thm 9.2 rank -> step

In the second sentence of def 9.4, the existence of the orthonormal
frame is automatic.

 

Now, besides some of the typos, the report is simply crap:

  • the referee complains that I’m doing it for groupoids, then says that what I am doing applies only to subriemannian spaces.
  • before, he says that in fact I’m doing it only for riemannian spaces.
  • I never claim that there is a main result in this long article, but somehow the referee mentions one of the theorems as the main result, while I am using it only as an example showing what the theory says in the trivial case, the one of riemannian manifolds.
  • the referee says that I don’t treat the sub-riemannian case. Should decide which is true, among the various claims, but take a look at the contents to get an opinion.
  • I never claim what the referee thinks are my two claims, both being of course wrong,
  • in the claim 1) (of the referee) he does not understand that the problem is not the definition of an operation, but the proof that the operation is a Carnot group one (I pass the whole story that in fact the operation is a conical group one, for regular sub-riemannian manifolds this translates into a Carnot group operation by using Siebert, too subtle for the referee)
  • the claim 2) is self-contradictory just by reading only the report.
  • 53 definitions (it is a very dense course), 15 theorems and 27 proofs, which are with no argument: “ not clear, incomplete, or totally obvious
  • but he goes on hunting the typos, thanks, that’s essential to show that he did read the article.

There is a part of the text which is especially perverse: The paper is a cut-and-past process from previous papers of the
author.

Mind you, this is a course based on several papers, most of them unpublished! Moreover, every contribution from previous papers is mentioned.

Tell me what to do with these papers: being unpublished, can I use them for a paper submitted to publication? Or else, they can be safely ignored because they are not published? Hmm.

This shows to me that the referee knows what I am doing, but he does not like it.

Fortunately, all the papers, published or not, are available on the arXiv with the submission dates and versions.

 

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See also previous posts:

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A less understood problem in sub-riemannian geometry (I)

A complete, locally compact riemannian manifold is a length metric space by the Hopf-Rinow theorem. The problem of intrinsic characterization of riemannian spaces asks for the recovery of the manifold structure and of the riemannian metric from the distance function coming from  to the length functional.

For 2-dim riemannian manifolds the problem has been solved by A. Wald in 1935. In 1948 A.D. Alexandrov  introduces his 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 Hölder smoothness).

The problem has been solved by Nikolaev in 1998, in the paper A metric characterization of Riemannian spaces. Siberian Adv. Math.   9,  no. (1999),  1-58.  The solution of Nikolaev can be summarized  like this: he starts with a locally compact length metric space (and some technical details), then

  •  he constructs a (family of) intrinsically defined tangent bundle(s) of the metric space, by using a generalization of the cosine formula for estimating a kind of a distance between two curves emanating from different points. This will lead him to a generalization of the tangent bundle of a riemannian manifold endowed with the canonical Sasaki metric.
  • He defines a notion of sectional curvature at a point of the metric space, as a limit of a function of nondegenerated geodesic triangles, limit taken as these triangles converge (in a precised sense)  to the point.
  • The sectional curvature function thus constructed is supposed to satisfy a Hölder continuity condition (thus a regularity formulated in metric terms)
  • He proves then that  the metric space is isometric with (the metric space associated to) a riemannian manifold of precise (weak) regularity (the regularity is related to the regularity of the sectional curvature function).

Sub-riemannian spaces are length metric spaces as well. Any riemannian space is a sub-riemannian one. It is not clear at first sight why the characterization of riemannian spaces does not extend to sub-riemannian ones. In fact, there are two problematic steps for such a program for extending Nikolaev result to sub-riemannian spaces:

  • the cosine formula, as well as the Sasaki metric on the tangent bundle don’t have a correspondent in sub-riemannian geometry (because there is, basically, no statement canonically corresponding to Pythagoras theorem);
  • the sectional curvature at a point cannot be introduced by means of comparison triangles, because sub-riemanian spaces do not behave well with respect to this comparison of triangle idea, as proved by Scott Pauls.

In 1996 M. Gromov formulates the problem of intrinsic characterization of sub-riemannian spaces.  He takes the Carnot-Caratheodory (or CC) distance (this is the name of the distance constructed on a sub-riemannian manifold from the differential geometric data we have, which generalizes the construction of the riemannian distance from the riemannian metric) as the only intrinsic object of a sub-riemannian space. Indeed, in the linked article, section 0.2.B. he writes:

If we live inside a Carnot-Caratheodory metric space V we may know nothing whatsoever about the (external) infinitesimal structures (i.e. the smooth structure on V, the subbundle H \subset T(V) and the metric g on H) which were involved in the construction of the CC metric.
He then formulates the goal:
Develop a sufficiently rich and robust internal CC language which would enable us to capture the essential external characteristics of our CC spaces.
He proposes as an example to recognize the rank of the horizontal distribution, but in my opinion this is, say, something much less essential than to “recognize” the “differential structure”, in the sense proposed here as the equivalence class under local equivalence of dilation structures.
As in Nikolaev solution for the riemannian case, the first step towards the goal is to have a well defined, intrinsic, notion of tangent bundle. The second step would be to be able to go to higher order approximations, eventually towards a curvature.
My solution is to base all on dilation structures. The solution is not “pure”, because it introduces another ingredient, besides the CC distance: the field of dilations. However, I believe that it is illusory to think that, for the general sub-riemannian case, we may be able to get a “sufficiently rich and robust” language without. As an example, even the best known thing, i.e. the fact that the metric tangent spaces of a (regular) sub-riemannian manifold are Carnot groups, was previously not known to be an intrinsic fact. Let me explain: all proofs, excepting the one by using dilation structures, use non-intrinsic ingredients, like differential calculus on the differential manifold which enters in the construction of the CC distance. Therefore, it is not known (or it was not known, even not understood as a problem) if this result is intrinsic or if it is an artifact of the proof method.
Well, it is not, it turns out, if we accept dilation structures as intrinsic.
There is a bigger question lingering behind, once we are ready to think about intrinsic properties of sub-riemannian spaces:  what is a sub-riemannian space? The construction of such spaces uses notions and results which are by no means intrinsic (again differential structures, horizontal bundles, and so on).
Therefore I understand Gromov’s stated goal as:
Give a minimal, axiomatic, description of sub-riemannian spaces.
[Adapted from the course notes Sub-riemannian geometry from intrinsic viewpoint.]

Dictionary from emergent algebra to graphic lambda calculus (I)

Because I am going to explore in future posts the emergent algebra sector, I think it is good to know where we stand with using graphic lambda calculus for describing proofs in emergent algebra as computations.  In the big map of research paths, this correspond to the black path linking “Energent algebra sector” with “Emergent algebras”.

A dictionary seems a good way to start this discussion.

Let’s see, there are three formalisms there:

  • in the first paper on spaces with dilations, Dilatation structures I. Fundamentals arXiv:math/0608536  section 4,  is introduced a formalism using binary decorated trees in order to ease the manipulations of dilation structures,
  • emergent algebras are an abstraction of dilation structures, in the sense that they don’t need a metric space to live on. The first paper on the subject is Emergent algebras arXiv:0907.1520   (see also Braided spaces with dilations and sub-riemannian symmetric spaces  arXiv:1005.5031  for explanations of the connection between dilation structures and emergent algebras, as well as for braided symmetric spaces, sub-riemannian symmetric spaces, conical groups, items you can see on the big map mentioned before). Emergent algebras is a mixture of an algebraic theory with an important part of epsilon-delta analysis.  One of the goals of graphic lambda calculus is to replace this epsilon-delta part by a computational part.
  • graphic lambda calculus, extensively described here, has an emergent algebra sector (see  arXiv:1305.5786 , equally check out the series Emergent algebras as combinatory logic part I, part II, part IIIpart IV,  ). This is not an algebraic theory, but a formalism which contains lambda calculus.

The first figure describes a dictionary of objects which appear in these three formalisms. In the first column you find objects as they appear in dilation structures – emergent algebra formalism. In the second column you find the corresponding object in the binary trees formalism. In the third column there are the respective objects as they appear in the emergent algebra sector of the graphic lambda calculus.

sum_basic

Some comments: the first two rows  are about an object called

  • “dilation (of coefficient \varepsilon, with \varepsilon \in \Gamma, a commutative group)” in dilation structures,  which is a operation in emergent algebras, indexed by \varepsilon, (the second row is about dilations of coefficient \varepsilon^{-1}),
  • it is an elementary binary tree with the node decorated by white (for \varepsilon) or black (for \varepsilon^{-1})
  • it is one of the elementary gates in graphic lambda calculus.

The third row is about the “approximate sum” in dilation structures, which is a composite operation in emergent algebras, which is a certain graph in graphic lambda calculus.

The fourth row is about the “approximate difference” and the fifth about the “approximate inverse”.

For the geometric meaning of these objects see the series on  The origin of emergent algebras part I, part II, part III,   or go directly and read arXiv:1304.3694 .

What is different between these rows?

  • In the first row we have an algebra structure based on identities between composites of operations defined on a set.
  • In the second row we have trees with leaves decorated by labels from an alphabet (of formal variables) or terms constructed recursively from those  .
  • In the third row we have graphs with no variable names. (Recall that in graphic lambda calculus there are no variable names. Everything written in red can be safely erased, it is put there only for the convenience of the reader.)

Let’s see now the dictionary of identities/moves.

moves_basic

The most important comment is that identities in emergent algebras become moves in the other two formalisms. A succession of moves is in fact a proof for an identity.

The names of the identities or moves have been commented in many places, you see there names like “Reidemeister move” which show relations to knot diagrams, etc. See this post for the names of the moves and relations to knot diagrams, as well as section 6 from  arXiv:1305.5786 .

Let’s read the first column: it says that from an algebraic viewpoint an emergent algebra is a one parameter family (indexed by \varepsilon \in \Gamma) of idempotent right quasigroups. From the geometric point of view of dilation structures, it is a formalisation of properties expected from an object called “dilation”, namely that it preserves the base-point ( “x” in the figure), that a composition of dilations of coefficients \varepsilon, \mu, with the same base-point,  is again a dilation, of coefficient \varepsilon \mu, etc.

In the second column we see two moves, R1 and R2, which can be applied anywhere in a decorated binary tree, as indicated.

In the third column we see that these moves are among the moves from graphic lambda calculus, namely that R1 is in fact related to the oriented Reidemeister move R1a, so it has the same name.

The fact that the idempotent right quasigroup indexed by the neutral element of \Gamma, denoted by 1, is trivial, has no correspondent for binary trees, but it appears as the move (ext 2) in graphic lambda calculus. Through the intermediary of this move appears the univalent termination gate.

These are the common moves. To these moves add, for the part of the emergent algebra sector, the R1b move, the local fan-out moves and some pruning moves. There is also the global fan-out move which is needed, but we are going to replace it by a local move which has the funny name of “linearity of fan-out”, but that’s for later.

The local fan-out moves and the pruning moves are needed for the emergent algebra sector but not for the binary trees or emergent algebras. They are the price we have to pay for eliminating variable names. See the algorithm for producing graphs from lambda calculus terms, for what concerns their use for solving the same problem for untyped lambda calculus. (However, the emergent algebra sector is to be compared not with the lambda calculus sector, but with the combinatory logic sector, more about this in a further post.)

We don’t need all pruning moves, but only one which, together with the local fan-out moves, form a family which could be aptly called:

co_mono_y

(notice I consider a reversible local pruning)

Grouping moves like this makes a nice symmetry with the fact that \Gamma is a commutative group, as remarked here.

As concerns the R1b move, which is the one from the next figure, I shall use it only if really needed (for the moment I don’t). It is needed for the knot diagrams made by emergent algebra gates sector, but it is not yet clear to me if we need it for the emergent algebra sector.

r1bmove

However, there is a correspondent of this move for emergent algebras. Indeed, recall that a right quasigroup is a a quasigroup if the equation x \circ a = b  has a solution, which is unique. If our emergent algebra is in fact a (family of) quasigroup(s) , as happens for the cases of conical groups or for symmetric spaces in the sense of Loos (explained in arXiv:1005.5031 ), then in particular it follows that the equation  x \circ_{\varepsilon} a = a has only the solution x = a (for \varepsilon \not = 1). This last statement has the R1b move as a correspondent in the realm of the emergent algebra sector.

Until now we have only local moves in the emergent algebra sector. We shall see that we need a global move (the global fan-out) in order to prove that the dictionary works, i.e. for proving the fundamental identities of emergent algebras within the graphic lambda calculus formalism. The goal will be to replace the global fan-out move by a new local move (i.e. one which is not a consequence of the existing moves of graphic lambda calculus). This new move will turn out to be a familiar sight, because it is related to the way we see linearity in emergent algebras.

Curvature and halfbrackets, part III

I continue from “Curvature and halfbrackets, part II“.  This post is dedicated to the application of the previously introduced notions to the case of a sub-riemannian Lie group.

_______________

1. I start with the definition of a sub-riemannian Lie group. If you look in the literature, the first reference to “sub-riemannian Lie groups” which I am aware about is the series Sub-riemannian geometry and Lie groups  arXiv:math/0210189, part II arXiv:math/0307342 , part III arXiv:math/0407099 .    However, that work predates the introduction of dilation structures, therefore there is a need to properly define this  object within the actual state of matters.

Definition 1. A sub-riemannian Lie group is a locally compact topological group G with the following supplementary structure:

  • together with the dilation structure coming from it’s one-parameter groups (by the Montgomery-Zippin construction), it has a group norm which induce a tempered dilation structure,
  • it has a left-invariant dilation structure (with dilations \delta^{x}_{\varepsilon} y = x \delta_{\varepsilon}(x^{-1}y) and group norm denoted by \| x \|) which, paired with the tempered dilation structure mentioned previously, it satisfies the hypothesis of “Sub-riemannian geometry from intrinsic viewpoint” Theorem 12.9,  arXiv:1206.3093

Remarks:

  1. there is no assumption on the tempered group norm to come from a Riemannian left-invariant distance on the group. For this reason, some people use the name sub-finsler  arXiv:1204.1613  instead of sub-riemannian, but I believe this is not a serious distinction, because the structure of a scalar product which induces the distance is simply not needed for understanding  sub-riemannian Lie groups.
  2. by Theorem 12.9, it follows that the left-invariant field of dilations induces a length dilation structure. I shall use this further. Length dilation structures are maybe a more useful object than simply dilation structures, because they explain how the length functional behaves at different scales, which is a much more detailed information about the microscopic structure of a length metric space than just the information about how the distance behaves at different scales.

This definition looks a bit mysterious, unless you read the course notes cited inside the definition. Probably, when I shall find the interest to pursue it, it would be really useful to just apply, step by step, the constructions from arXiv:1206.3093 to sub-riemannian Lie groups.

__________________

2. With the notations from the last post, I want to compute the quantities A, B, C. We already know that B is related to the curvature of G with respect to it’s sub-riemannian (sub-finsler if you like it more) distance, as introduced previously via metric profiles.  We also know that B is controlled by A and C. But let’s see the expressions of these three quantities for sub-riemannian Lie groups.

I denote by d(u,v) the left invariant sub-riemannian distance, therefore we have d(u,v) = \| u^{-1}v\|.

Now, \rho_{\varepsilon}(x,u) = \| x^{-1} u \|_{\varepsilon} , where \varepsilon \| u \|_{\varepsilon} = \| \delta_{\varepsilon} u \|  by definition.  Notice also that \Delta^{x}_{\varepsilon}(u,v) = (\delta^{x}_{\varepsilon} u ) ((u^{-1} x) *_{\varepsilon} (x^{-1} v)), where  u *_{\varepsilon} v is the deformed group operation at scale \varepsilon, i.e. it is defined by the relation:

\delta_{\varepsilon} (u *_{\varepsilon} v) = (\delta_{\varepsilon} u) (\delta_{\varepsilon} v)

With all this, it follows that:

A_{\varepsilon}(x,u) = \rho_{\varepsilon}(x,u) - d^{x}(x,u) = \|x^{-1} u \|_{\varepsilon} - \| x^{-1} u \|_{0}

A_{\varepsilon}(\delta^{x}_{\varepsilon} u, \Delta^{x}_{\varepsilon}(u,v)) = \| (u^{-1} x) *_{\varepsilon} (x^{-1} v) \|_{\varepsilon} - \| (u^{-1} x) *_{\varepsilon} (x^{-1} v)\|_{0}.

A similar computation leads us to the expression for the curvature related quantity

B_{\varepsilon}(x,u,v) = d^{x}_{\varepsilon}(u,v) - d^{x}(u,v) = \| (u^{-1}x) *_{\varepsilon} (x^{-1} v)\|_{\varepsilon} - \| (u^{-1}x) *_{0} (x^{-1}v)\|_{0}.

Finally,

C_{\varepsilon}(x,u,v) = \|(u^{-1} x) *_{\varepsilon} (x^{-1} v)\|_{0} - \|(u^{-1}x) *_{0} (x^{-1}v)\|_{0}. This last quantity is controlled by a halfbracket, via a norm inequality.

The expressions of A, B, C make transparent that the curvature-related B is the sum of A and C. In the next post I shall use the length dilation structure of the sub-riemannian Lie group in order to show that A is controlled by C, which in turn is controlled by a norm of a halfbracket. Then I shall apply all this to SO(3), as an example.

Curvature and halfbrackets, part II

I continue from “Curvature and halfbrackets, part I“, with the same notations and background.

In a metric space with dilations (X,d,\delta), there are three quantities which will play a role further.

1. The first quantity is related to the “norm” function defined as

\rho_{\varepsilon}(x,u) = d^{x}_{\varepsilon}(x,u)

Notice that this is not a distance function, instead it is more like a norm of u with respect to the basepoint x, at scale \varepsilon. Together with the field of dilations, this “norm” function contains all the information about the local and infinitesimal behaviour of the distance d. We can see this from the fact that we can recover the re-scaled distance d^{x}_{\varepsilon} from this “norm”, with the help of the approximate difference (for this notion see on this blog the definition of approximate difference in terms of emergent algebras here, or go to point 3. from the post The origin of emergent algebras (part III)):

\rho_{\varepsilon}(\delta^{x}_{\varepsilon} u , \Delta^{x}_{\varepsilon}(u,v)) = d^{x}_{\varepsilon}(u,v)

(proof left to the interested reader) This identity shows that the uniform convergence of (x,u,v) \mapsto d^{x}_{\varepsilon}(u,v) to (x,u,v) \mapsto d^{x}(u,v), as \varepsilon goes to 0, is a consequence of the following pair of uniform convergences:

  • that of the function (x,u) \mapsto \rho_{\varepsilon}(x,u) which converges to (x,u) \mapsto d^{x}(x,u)
  • that of  the pair (dilation, approximate difference)  (x,u,v) \mapsto (\delta^{x}_{\varepsilon} u , \Delta^{x}_{\varepsilon}(u,v)) to (x,u,v) \mapsto (x, \Delta^{x}(u,v)), see how this pair appears from the normed groupoid formalism, for example by reading the post from the post The origin of emergent algebras (part III).

With this definition of the “norm” function, I can now introduce the first quantity of interest, which measures the difference between the “norm” function at scale \varepsilon and the “norm” function at scale 0:

A_{\varepsilon}(x,u) = \rho_{\varepsilon}(x,u) - d^{x}(x,u)

The interpretation of this quantity is easy in the particular case of a riemannian space with dilations defined by the geodesic exponentials. In this particular case

A_{\varepsilon}(x,u) = 0

because the “norm” function \rho_{\varepsilon}(x,u) equals the distance between d(x,u) (due to the definition of dilations with respect to the geodesic exponential).

In more general situations, for example in the case of a regular sub-riemannian space, we can’t define dilations in terms of geodesic exponentials (even if we may have at disposal geodesic exponentials). The reason has to do with the fact that the geodesic exponential in the case of a regular sub-riemannian manifold, is not intrinsically defined as a function from the tangent of the geodesic at it’s starting point. That is because geodesics in regular sub-riemannian manifolds (at least those which are classically, i.e. with respect to the differential manifold structure, smooth , are bound to have tangents only in the horizontal directions.

As another example, think about a sub-riemannian Lie group. Here, we may define a left-invariant dilation structure with the help of the Lie group exponential. In this case the quantity A_{\varepsilon}(x,u) is certainly not equal to 0, excepting very particular cases, as a riemannian compact Lie group, with bi-invariant distance, where the geodesic and Lie group exponentials coincide.

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2.   The second quantity is the one which is most interesting for defining (sectional like) curvature, let’s call it

B_{\varepsilon}(x,u,v) = d^{x}_{\varepsilon}(u,v) - d^{x}(u,v).

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3. Finally, the third quantity of interest is a kind of a measure of the convergence of (x,u,v) \mapsto (\delta^{x}_{\varepsilon} u , \Delta^{x}_{\varepsilon}(u,v)) to (x,u,v) \mapsto (x, \Delta^{x}(u,v)), but measured with the norms from the tangent spaces.  Now, a bit of notations:

dif_{\varepsilon}(x,u,v) = (\delta^{x}_{\varepsilon} u , \Delta^{x}_{\varepsilon}(u,v)) for any three points x, u, v,

dif_{0}(x,u,v) = (x, \Delta^{x}(u,v))  for any three points x, u, v  and

g(v,w) = d^{v}(v,w) for any two points v, w.

With these notations I introduce the third quantity:

C_{\varepsilon}(x,u,v) = g( dif_{\varepsilon}(x,u,v) ) - g( dif_{0}(x,u,v) ).

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The relation between these three quantities is the following:

Proposition.  B_{\varepsilon}(x,u,v) = A_{\varepsilon}(dif_{\varepsilon}(x,u,v)) + C_{\varepsilon}(x,u,v).

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Suppose that we know the following estimates:

A_{\varepsilon}(x,u) = \varepsilon^{\alpha} A(x,u) + higher order terms, with A(x, u) \not = 0 and \alpha > 0,

B_{\varepsilon}(x,u,v) = \varepsilon^{\beta} B(x,u,v) + higher order terms, with B(x,u,v) \not = 0 and \beta > 0,

C_{\varepsilon}(x,u) = \varepsilon^{\gamma} C(x,u,v) + higher order terms, with C(x, u) \not = 0 and \gamma > 0,

Lemma. Let  us sort in increasing order the list of the values \alpha, \beta, \gamma and denote the sorted list by a, b, c. Then a = b.

The proof is easy. The equality from the Proposition tells us that the modules of A_{\varepsilon}, B_{\varepsilon} and C_{\varepsilon} can be taken as the edges of a triangle. Suppose then that a < b < c, use the estimates from the hypothesis and divide by \varepsilon^{a} in one of the three triangle inequalities, then go with \varepsilon to 0 in order to arrive at a contradiction 0 < 0.

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The moral of the lemma is that there are at most two different coefficients in the list \alpha, \beta, \gamma. The coefficient \beta is called “curvdimension”. In the next post I shall explain why, in the case of a sub-riemannian Lie group,  the coefficient \gamma is related to the halfbracket. Moreover, we shall see that in the case of sub-riemannian Lie groups all three coefficient are equal, therefore the infinitesimal behaviour of the halfbracket determines the curvdimension.

Emergent algebras as combinatory logic (Part II)

This post continues Emergent algebras as combinatory logic (Part I).  My purpose is to introduce the calculus standing behind Theorem 1 from the mentioned post.

We have seen (Definition 2) that there are approximate sum and difference operations associated to an emergent algebra. Let me add to them a third operation, namely the approximate inverse. For clarity I repeat here the Definition 2, supplementing it with the definition of the approximate inverse. This gives:

Definition 2′.   For any \varepsilon \in \Gamma we give the following names to several combinations of operations of emergent algebras:

  • the approximate sum operation is \Sigma^{x}_{\varepsilon} (u,v) = x \bullet_{\varepsilon} ((x \circ_{\varepsilon} u) \circ_{\varepsilon} v),
  • the approximate difference operation is \Delta^{x}_{\varepsilon} (u,v) = (x \circ_{\varepsilon} u) \bullet_{\varepsilon} (x \circ_{\varepsilon} v)
  • the approximate inverse operation is inv^{x}_{\varepsilon} u = (x \circ_{\varepsilon} u) \bullet_{\varepsilon} x.

The justification for these names comes from the explanations given in the post “The origin of emergent algebras (part II)“, where I discussed the sketch of a solution to the question “What makes the (metric)  tangent space (to a sub-riemannian regular manifold) a group?”, given by Bellaiche in the last two sections of his article  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. We have seen there that the group operation (the noncommutative,  in principle, addition of vectors) can be seen as the limit of compositions of intrinsic dilations, as \varepsilon goes to 0. It is important that this limit exists and that it is uniform, according to Gromov’s hint.

Well,  with the notation \delta^{x}_{\varepsilon} y = x \circ_{\varepsilon} y, \delta^{x}_{\varepsilon^{-1}} y = x \bullet_{\varepsilon} y, it becomes clear, for example, that the composition of intrinsic dilations described in the figure from the post “The origin of emergent algebras (part II)” is nothing but the approximate sum from Definition 2′. (This is to say that formally, if we replace the emergent algebra operations with the respective intrinsic dilations, then the approximate sum operation \Sigma^{x}_{\varepsilon}(y,z)  appears as the red point E from the mentioned  figure. It is still left to prove that intrinsic dilations from regular sub-riemannian spaces give rise to emergent algebras, this was done in arXiv:0708.4298.)

We recognize therefore the two ingredients of Bellaiche’s solution into the definition of an emergent algebra:

  • approximate operations, which are just clever compositions of intrinsic dilations in the realm of sub-riemannian spaces, which
  • converge in a uniform way to the exact operations which give the algebraic structure of the tangent space.

Therefore, a rigorous formulation of Bellaiche’s solution is Theorem 1 from the previous post, provided that we extract,  from the long differential geometric work done by Bellaiche, only the part which is necessary for proving that intrinsic dilations produce an emergent algebra structure.

Nevertheless, Theorem 1 shows that the “emergence of operations” phenomenon is not at all specific to sub-riemannian geometry. In fact, once we get the idea of the right definition of approximate operations (from sub-riemannian geometry), we can simply try to prove the theorem by “abstract nonsense”, i.e. algebraically, with a dash of uniform convergence at the end.

For this we have to identify the algebraic relations which are satisfied by these approximate operations.  For example, is the approximate sum associative? is the approximate difference the inverse of the approximate sum? is the approximate inverse of an element the inverse with respect to the approximate sum? and so on. The answer to these questions is “approximately yes”.

It is clear that in order to find the right relations (approximate associativity and so on) between these approximate operations we need to reason in a more clear way. Just by looking at the expressions of the operations from Definition 2′, it is obvious that if we start with a brute force  “shut up and compute” approach  then we will end rather quickly with a mess of parantheses and coefficients. There has to be a more easy way to deal with those approximate operations than brute force.

The way I have found has to do with a graphical representation of these operations, a way which eventually led me to graphic lambda calculus. This is for next time.