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.


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