A less understood problem in sub-riemannian geometry (I)

A complete, locally compact riemannian manifold is a length metric space by the Hopf-Rinow theorem. The problem of intrinsic characterization of riemannian spaces asks for the recovery of the manifold structure and of the riemannian metric from the distance function coming from  to the length functional.

For 2-dim riemannian manifolds the problem has been solved by A. Wald in 1935. In 1948 A.D. Alexandrov  introduces his 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 Hölder smoothness).

The problem has been solved by Nikolaev in 1998, in the paper A metric characterization of Riemannian spaces. Siberian Adv. Math.   9,  no. (1999),  1-58.  The solution of Nikolaev can be summarized  like this: he starts with a locally compact length metric space (and some technical details), then

  •  he constructs a (family of) intrinsically defined tangent bundle(s) of the metric space, by using a generalization of the cosine formula for estimating a kind of a distance between two curves emanating from different points. This will lead him to a generalization of the tangent bundle of a riemannian manifold endowed with the canonical Sasaki metric.
  • He defines a notion of sectional curvature at a point of the metric space, as a limit of a function of nondegenerated geodesic triangles, limit taken as these triangles converge (in a precised sense)  to the point.
  • The sectional curvature function thus constructed is supposed to satisfy a Hölder continuity condition (thus a regularity formulated in metric terms)
  • He proves then that  the metric space is isometric with (the metric space associated to) a riemannian manifold of precise (weak) regularity (the regularity is related to the regularity of the sectional curvature function).

Sub-riemannian spaces are length metric spaces as well. Any riemannian space is a sub-riemannian one. It is not clear at first sight why the characterization of riemannian spaces does not extend to sub-riemannian ones. In fact, there are two problematic steps for such a program for extending Nikolaev result to sub-riemannian spaces:

  • the cosine formula, as well as the Sasaki metric on the tangent bundle don’t have a correspondent in sub-riemannian geometry (because there is, basically, no statement canonically corresponding to Pythagoras theorem);
  • the sectional curvature at a point cannot be introduced by means of comparison triangles, because sub-riemanian spaces do not behave well with respect to this comparison of triangle idea, as proved by Scott Pauls.

In 1996 M. Gromov formulates the problem of intrinsic characterization of sub-riemannian spaces.  He takes the Carnot-Caratheodory (or CC) distance (this is the name of the distance constructed on a sub-riemannian manifold from the differential geometric data we have, which generalizes the construction of the riemannian distance from the riemannian metric) as the only intrinsic object of a sub-riemannian space. Indeed, in the linked article, section 0.2.B. he writes:

If we live inside a Carnot-Caratheodory metric space V we may know nothing whatsoever about the (external) infinitesimal structures (i.e. the smooth structure on V, the subbundle H \subset T(V) and the metric g on H) which were involved in the construction of the CC metric.
He then formulates the goal:
Develop a sufficiently rich and robust internal CC language which would enable us to capture the essential external characteristics of our CC spaces.
He proposes as an example to recognize the rank of the horizontal distribution, but in my opinion this is, say, something much less essential than to “recognize” the “differential structure”, in the sense proposed here as the equivalence class under local equivalence of dilation structures.
As in Nikolaev solution for the riemannian case, the first step towards the goal is to have a well defined, intrinsic, notion of tangent bundle. The second step would be to be able to go to higher order approximations, eventually towards a curvature.
My solution is to base all on dilation structures. The solution is not “pure”, because it introduces another ingredient, besides the CC distance: the field of dilations. However, I believe that it is illusory to think that, for the general sub-riemannian case, we may be able to get a “sufficiently rich and robust” language without. As an example, even the best known thing, i.e. the fact that the metric tangent spaces of a (regular) sub-riemannian manifold are Carnot groups, was previously not known to be an intrinsic fact. Let me explain: all proofs, excepting the one by using dilation structures, use non-intrinsic ingredients, like differential calculus on the differential manifold which enters in the construction of the CC distance. Therefore, it is not known (or it was not known, even not understood as a problem) if this result is intrinsic or if it is an artifact of the proof method.
Well, it is not, it turns out, if we accept dilation structures as intrinsic.
There is a bigger question lingering behind, once we are ready to think about intrinsic properties of sub-riemannian spaces:  what is a sub-riemannian space? The construction of such spaces uses notions and results which are by no means intrinsic (again differential structures, horizontal bundles, and so on).
Therefore I understand Gromov’s stated goal as:
Give a minimal, axiomatic, description of sub-riemannian spaces.
[Adapted from the course notes Sub-riemannian geometry from intrinsic viewpoint.]
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