This is a talk in the Conference on Physical Knotting, Vortices and Surgery in Nature.
This is a talk in the Conference on Physical Knotting, Vortices and Surgery in Nature.
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 with the following supplementary structure:
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 . We already know that is related to the curvature of 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 is controlled by and . But let’s see the expressions of these three quantities for sub-riemannian Lie groups.
I denote by the left invariant sub-riemannian distance, therefore we have .
Now, , where by definition. Notice also that , where is the deformed group operation at scale , i.e. it is defined by the relation:
With all this, it follows that:
A similar computation leads us to the expression for the curvature related quantity
. This last quantity is controlled by a halfbracket, via a norm inequality.
The expressions of make transparent that the curvature-related is the sum of and . In the next post I shall use the length dilation structure of the sub-riemannian Lie group in order to show that is controlled by , which in turn is controlled by a norm of a halfbracket. Then I shall apply all this to , as an example.
I continue from “Curvature and halfbrackets, part I“, with the same notations and background.
In a metric space with dilations , there are three quantities which will play a role further.
1. The first quantity is related to the “norm” function defined as
Notice that this is not a distance function, instead it is more like a norm of with respect to the basepoint , at scale . Together with the field of dilations, this “norm” function contains all the information about the local and infinitesimal behaviour of the distance . We can see this from the fact that we can recover the re-scaled distance 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)):
(proof left to the interested reader) This identity shows that the uniform convergence of to , as goes to , is a consequence of the following pair of uniform convergences:
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 and the “norm” function at scale :
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
because the “norm” function equals the distance between (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 is certainly not equal to , excepting very particular cases, as a riemannian compact Lie group, with bi-invariant distance, where the geodesic and Lie group exponentials coincide.
2. The second quantity is the one which is most interesting for defining (sectional like) curvature, let’s call it
3. Finally, the third quantity of interest is a kind of a measure of the convergence of to , but measured with the norms from the tangent spaces. Now, a bit of notations:
for any three points ,
for any three points and
for any two points .
With these notations I introduce the third quantity:
The relation between these three quantities is the following:
Suppose that we know the following estimates:
higher order terms, with and ,
higher order terms, with and ,
higher order terms, with and ,
Lemma. Let us sort in increasing order the list of the values and denote the sorted list by . Then .
The proof is easy. The equality from the Proposition tells us that the modules of , and can be taken as the edges of a triangle. Suppose then that , use the estimates from the hypothesis and divide by in one of the three triangle inequalities, then go with to in order to arrive at a contradiction .
The moral of the lemma is that there are at most two different coefficients in the list . The coefficient is called “curvdimension”. In the next post I shall explain why, in the case of a sub-riemannian Lie group, the coefficient 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.
Here is a little story about curvature and Lie brackets in the wider context of dilation structures and sub-riemannian Lie groups. The background of this post is provided by the following links:
We are in the frame of a metric space with dilations (aka “dilation structure” or “dilatation structure”). I introduced these spaces under the name “dilatation structures” in the article Dilatation structures I. Fundamentals, arXiv:math/0608536 [math.MG], but see the course notes for the most advanced formulation. In particular regular sub-riemannian manifolds, riemannian manifolds and Lie groups with a left invariant distance induced by a completely non-integrable (i.e. generating) distribution are examples of such spaces.
There are two problems, both without a clear solution yet, concerning this class of spaces:
My purpose is to explain a link between the curvature (defined as the curvature of a metric profile) and the halfbracket.
With the notations from dilation structures, we know that for any sufficiently close and any sufficiently small we have the uniform estimate
Here is the distance between and , seen as elements of the tangent space at . The quantity
is the deformation of the distance by dilations , centered at , of coefficient .
Let’s take, as an example, the case of a riemannian manifold with geodesic exponential and dilations defined by:
In this case
where is related to the sectional curvature at (we suppose that are not collinear). This gives a curvdimension equal to and also a notion of sectional curvature.
But in general all we can hope is an estimate of the form
where is the curvdimension, and the trick is to have an estimate for the curvdimension. In the next post I shall use halfbrackets in order to estimate the curvdimension, for sub-riemannian Lie groups.
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 riemannian spaces:
every locally compact length metric space , not linear at one of its points, with Holder continuous metric sectional curvature of the “generalized tangent bundle” (for some $m=1,2,…$, which admits local geodesic extendability, is isometric to a smooth riemannian manifold..
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.
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. , the equivalent of 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, 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.
I continue from the previous post “Curvdimension and curvature of a metric profile II“.
Let’s see what is happening for , a sufficiently smooth ( for example), complete, connected riemannian manifold. The letter “” denotes the metric (scalar product on the tangent space) and the letter “” will denote the riemannian distance, that is for any two points the distance between them is the infimum of the length of absolutely continuous curves which start from and end in . The length of curves is computed with the help of the metric .
Notations. In this example 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 be vectors in the tangent space at $x \in X$. When the basepoint is fixed by the context then I may renounce to mention it in the various notations. For example means the norm of the vector with respect to the scalar product on the tangent space at the point . Likewise, may be used instead of ; the riemannian curvature tensor at may be denoted by and not by , and so on …
Remark 2. The smoothness of the riemannian manifold should be just enough such that the curvature tensor is and such that for any compact subset of , possibly by rescaling , the geodesic exponential makes sense (exists and it is uniquely defined) for any and for any with .
Let us fix such a compact set and let’s take a point .
Definition 5. For any we define on the closed ball of radius centered at (with respect to the distance ) the following distance: for any with ,
(The notation used here is in line with the one used in dilation structures.)
Recall that the sectional curvature is defined for any pair of vectors which are linearly independent (i.e. non collinear).
Proposition 1. Let be greater or equal than , for any and any non-collinear pair of vectors with , . Then for any and any , with , we have
Corollary 1. For any the metric space has a metric tangent space at , which is the isometry class of the unit ball in with the distance .
Corollary 2. If the sectional curvature at is non trivial then the metric profile at has curvdimension 2 and moreover
This continues the previous post Curvdimension and curvature of a metric profile, I.
Definition 3. (flat space) A locally compact metric space is locally flat around if there exists such that for any we have . A locally compact metric space is flat if the metric profile at any point is eventually constant.
Question 1. Metric tangent spaces are locally flat but are they locally flat everywhere? I don’t think so, but I don’t have an example.
Definition 4. Let be a locally compact metric space and a point where the metric space admits a metric tangent space. The curvdimension of at is , where is the set of all such that
Remark that the set always contains . Also, according to this definition, if the space is locally flat around then the curvdimension at is .
Question 2. Is there any metric space with infinite curvdimension at a point where the space is not locally flat? (Most likely the answer is “yes”, a possible example would be the revolution surface obtained from a graph of a infinitely differentiable function such that and all derivatives of at are equal to . This surface is taken with the distance from the 3-dimensional space, but maybe I am wrong. )
We are going to see next that the curvdimension of a sufficiently smooth riemannian manifold at any of its points where the sectional curvature is not trivial is equal to .
In the notes Sub-riemannian geometry from intrinsic viewpoint I propose two notions related to the curvature of a metric space at one of its points: the curvdimension and the curvature of a metric profile.In this post I would like to explain in detail what is this about, as well as making a number of comments and suggestions which are not in the actual version of the notes.
I shall start with the definition of the metric profile associated to a point of a locally compact metric space . We need first a short preparation.
Let be the collection of isometry classes of pointed compact metric spaces.An element of is denoted like and is the equivalence class of a compact metric space , with a specified point , with respect to the equivalence relation: two pointed compact metric spaces , are equivalent if there is a surjective isometry such that .
The space $CMS$ is a metric space when endowed with the Gromov-Hausdorff distance between (isometry classes of) pointed compact metric spaces.
Definition 1. Let be a locally compact metric space. The metric profile of at is the function which associates to the element of defined by
(defined for small enough , so that the closed metric ball with respect to the distance , is compact).
Remark 1. See the previous post Example: Gromov-Hausdorff distance and the Heisenberg group, part II , where the behaviour of the metric profile of the physicists Heisenberg group is discussed.
The metric profile of the space at a point is therefore a curve in another metric space, namely with a Gromov-Hausdorff distance. It is not any curve, but one which has certain properties which can be expresses with the help of the GH distance. Very intriguing, what about a dynamic induced along these curves in the . Nothing is known about this, strangely!
Indeed, to any element of it is associated the curve . This curve could be renamed . Notice that .
For a fixed , take now , what is the metric profile of this element of ? The answer is: for any we have
which proves that the curves in which are metric profiles are not just any curves.
Definition 2. If the metric profile can be extended by continuity to , then the space $(X,d)$ admits a metric tangent space at and the isometry class of (the unit ball in) the tangent space equals .
You see, cannot be any point from $CMS$. It has to be the isometry class of a metric cone, namely a point of which has constant metric profile.
The curvdimension and curvature explain how the the metric profile curve behaves near . This is for the next post.
The Brunn-Minkowski inequality says that the log of the volume (in euclidean spaces) is concave. The concavity inequality is improved, in riemannian manifolds with Ricci curvature at least K, by a quadratic term with coefficient proportional with K.
The paper is remarkable in many ways. In particular are compared two roads towards curvature in spaces more general than riemannian: the coarse curvature introduced by Ollivier and the other based on the displacement convexity of the entropy function (Felix Otto , Cedric Villani, John Lott, Karl-Theodor Sturm), studied by many researchers. Both are related to Wasserstein distances . NONE works for sub-riemannian spaces, which is very very interesting.
In few words, here is the description of the coarse Ricci curvature: take an epsilon and consider the application from the metric space (riemannian manifold, say) to the space of probabilities which associates to a point from the metric space the restriction of the volume measure on the epsilon-ball centered in that point (normalized to give a probability). If this application is Lipschitz with constant L(epsilon) (on the space of probabilities take the L^1 Wassertein distance) then the epsilon-coarse Ricci curvature times epsilon square is equal to 1 minus L(epsilon) (thus we get a lower bound of the Ricci curvature function, if we are in a Riemannian manifold). Same definition works in a discrete space (this time epsilon is fixed).
The second definition of Ricci curvature comes from inverse engineering of the displacement convexity inequality discovered in many particular spaces. The downside of this definition is that is hard to “compute” it.
Initially, this second definition was related to the L^2 Wasserstein distance which, according to Otto calculus, gives to the space of probabilities (in the L^2 frame) a structure of an infinite dimensional riemannian manifold.
Concerning the sub-riemannian spaces, in the first definition the said application cannot be Lipschitz and in the second definition there is (I think) a manifestation of the fact that we cannot put, in a metrically acceptable way, a sub-riemannian space into a riemannian-like one, even infinite dimensional.