Tag Archives: curvature

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


  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}.


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.


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).


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) ).


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).


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.


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.

Curvature and halfbrackets, part I

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:

  • Sub-riemannian geometry from intrinsic viewpoint, course notes, arXiv:1206.3093 [math.MG], especially section 2.5 “Curvdimension and curvature” and 12.3 “Coherent projections induce length dilation structures”,
  • Curvdimension and curvature of a metric profile part I, part II, part III,
  • Noncommutative Baker-Campbell-Haussdorf formula, the problem and a suggested solution.

We are in the frame of a metric space with dilations (X,d,\delta) (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:

  • Problem 1: how to define a good notion of curvature?  In particular, how to define an intrinsic notion of curvature for a regular sub-riemannian manifold, such that, for example, the curvature of a Carnot group is null (i.e. Carnot groups are flat)? On one side, I have a definition, the one about the curvature of a metric profile, which I think is good, but I have troubles computing it for particular examples. On the other side, all other notions of curvature, like for example those coming from the Wasserstein distance, fail spectacularly for sub-riemannian spaces.
  • Problem 2: for sub-riemannian Lie groups, or more general for groups with dilations  (i.e. left-invariant dilation structures on a topological group), how to define a good generalization of a Lie bracket? As I explained in the posts about the noncommutative BCH formula, this problem comes from the two ways we may interpret the Lie bracket. The first way, classical, says that the Lie bracket is an object which measures the noncommutativity of the group operation. This is in fact a statement which applies not to the Lie bracket, but to the commutator. The Lie bracket itself is related to the second order variation of the commutator (i.e.  the commutator of two elements of the group, which are \varepsilon-small, equals \varepsilon^{2} times the Lie bracket of those elements plus terms of higher order in \varepsilon. The trouble with this definition of the Lie bracket is that if we replace the “usual” dilations associated to a Lie group, those coming from one-parameter subgroups, by  more general dilations then all the reasoning crushes in at least two places. The first place is that in the tangent space at the identity of the group we have a noncommutative (but nilpotent) addition operation instead of the commutative plus operation, i.e. a Carnot group instead of a vector space structure, which makes hard to understand what sense the BCH formula makes. The second place is that instead of  the \varepsilon^{2}  term, there is nothing clear about the “expansion” of the commutator with respect to \varepsilon in this more general case.  Another interpretation of the Lie bracket comes from the fact that a half of the Lie bracket  measures the speed of the deformation of the group operation by dilations, at \varepsilon = 0. So, in this  interpretation, the half of the Lie bracket appears as the first order variation of the deformed group operation. This can be generalized to the more general case of a group with dilations and it leads to the notion of a halfbracket.

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 x, u, v sufficiently close and any \varepsilon > 0  sufficiently small we have the uniform estimate

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

Here d^{x}(u,v)  is the distance between u and v, seen as elements of the tangent space at x. The quantity

d^{x}_{\varepsilon}(u,v)\frac{1}{\varepsilon} d(\delta^{x}_{\varepsilon} u , \delta^{x}_{\varepsilon} v)

is the deformation of the distance d by dilations \delta^{x}_{\varepsilon}, centered at x, of coefficient \varepsilon.

Let’s take, as an example, the case of a riemannian manifold with geodesic exponential \exp and dilations defined by:

\delta^{x}_{\varepsilon} \exp_{x} u = \exp_{x} (\varepsilon u).

In this case

d^{x}_{\varepsilon}(u,v) - d^{x}(u,v) = \varepsilon^{2} (M + O(\varepsilon))

where M is related to the sectional curvature at x (we suppose that u, v are not collinear).  This gives a curvdimension equal to 2 and also a notion of sectional curvature.

But in general all we can hope is an estimate of the form

d^{x}_{\varepsilon}(u,v) - d^{x}(u,v) = \varepsilon^{\alpha} (M + O(\varepsilon))

where \alpha > 0 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.

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..


  • 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.

Curvdimension and curvature of a metric profile, II

This continues the previous post Curvdimension and curvature of a metric profile, I.

Definition 3. (flat space) A locally compact metric space (X,d) is locally flat around x \in X if there exists a > 0 such that for any \varepsilon, \mu \in (0,a] we have P^{m}(\varepsilon , [X,d,x]) = P^{m}(\mu , [X,d.x]). A locally compact metric space is flat if the metric profile at any point is eventually constant.

Carnot groups  and, more generally, normed conical groups are flat.

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 (X,d) be a  locally compact metric space and x \in X a point where the metric space admits a metric tangent space. The curvdimension of (X,d) at x is curvdim \, (X,d,x) = \sup M, where  M \subset [0,+\infty) is the set of all \alpha \geq 0 such that

\lim_{\varepsilon \rightarrow 0} \frac{1}{\varepsilon^{\alpha}} d_{GH}(P^{m}(\varepsilon , [X,d,x]) , P^{m}( 0 , [X,d,x])) = 0

Remark that the set M always contains 0. Also, according to this definition, if the space is locally flat around x then the curvdimension at x is + \infty.

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 f such that f(0) = 0 and all derivatives of f at 0 are equal to 0. 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 2.

Curvdimension and curvature of a metric profile, part I

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.

Related to these notions, they stem from rather vague proposals first made in earlier papers Curvature of sub-riemannian spaces and Sub-riemannian geometry and Lie groups II.

I shall start with the definition of the metric profile associated to a point x \in X of a locally compact metric space (X,d).  We need first a short preparation.

Let CMS be the collection of isometry classes of  pointed compact metric spaces.An element of CMS is denoted like [X,d,x] and is the equivalence class of a compact metric space (X,d), with a specified point x\in X, with respect to the equivalence relation: two pointed compact metric spaces (X,d,x), (Y,D,y) are equivalent if there is a surjective  isometry f: (X,d) \rightarrow (Y,D) such that f(x) = y.

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 (X,d) be a locally compact metric space. The metric profile of (X,d) at x is the function which associates to \varepsilon > 0 the element of CMS defined by

P^{m}(\varepsilon, x) = \left[\bar{B}(x,1), \frac{1}{\varepsilon} d, x\right]

(defined for small enough \varepsilon, so that the closed metric ball \bar{B}(x,\varepsilon) with respect to the distance d,  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 CMS 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 CMS. Nothing is known about this, strangely!

Indeed, to any element [X,d,x] of CMS it is associated the curve P^{m}(\varepsilon,x). This curve could be renamed P^{m}(\varepsilon , [X,d,x]).  Notice that P^{m}(1 , [X,d,x]) = [X,d,x].

For a fixed \varepsilon \in (0,1], take now P^{m}(\varepsilon , [X,d,x]), what is the metric profile of this element of CMS? The answer is: for any \mu \in (0,1] we have

P^{m}(\mu , P^{m}(\varepsilon , [X,d,x])) = P^{m}(\varepsilon \mu , [X,d,x])

which proves that the curves in CMS which are metric profiles are not just any curves.

Definition 2. If the metric profile P^{m}(\varepsilon ,[X,d,x]) can be extended by continuity to \varepsilon = 0, then the space $(X,d)$ admits a metric tangent space at x \in X and the isometry class of (the unit ball in) the tangent space equals  P^{m}(0 , [X,d,x]).

You see, P^{m}(0 , [X,d,x]) cannot be any point from $CMS$. It has to be the isometry class of a metric cone, namely a point of CMS which has constant metric profile.

The curvdimension and curvature explain how the the metric profile curve behaves near \varepsilon = 0. This is for the next post.

Curvature and Brunn-Minkowski inequality

A beautiful paper by Yann Ollivier and Cedric Villani


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.