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.

4 thoughts on “Curvdimension and curvature of a metric profile, II”

  1. “…are they locally flat everywhere?” What would act as a measure of the flatness of a metric tangent space? It seems that we would have to define another tangent metric space to define what ‘locally flat everywhere’ is. What about the set of endomorphisms of the metric tangent space onto itself?
    Is there a way to darken the text of the equations? It is so light that it is hard to read the equations. 😦

    1. I don’t know how to darken the equations.

      Metric tangent spaces are metric spaces, therefore definition 3 of flatness applies to them too. According to this definition, a metric tangent space is locally flat (around its basepoint), but is it locally flat around other points?

      Concerning the set of endomorphisms, metric tangent spaces are self-similar (w.r.t. the basepoint). In some cases, like for example when they are conical groups, they are globally flat, meaning that they are self-similar with respect to any of their points. This is a definition of “linearity”, albeit not a standard one. In this cases the group of endomorphisms is a (generalization of the) linear group. For example, the Heisenberg group is “linear” and its “linear group” is a semidirect product of (left) translations with conformally symplectic linear (in the usual sense) transformations.

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