# 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 $2$.