I expressed several times the belief that sub-riemannian geometry represents an example of a mathematically new phenomenon, which I call “non-commutative analysis”. Repeatedly happened that apparently general results simply don’t work well when applied to sub-riemannian geometry. This “strange” (not for me) phenomenon leads to negative statements, like rigidity results (Mostow, Margulis), non-rectifiability results (like for example the failure of the theory of metric currents for Carnot groups). And now, to this adds the following, arXiv:1404.5494 [math.OA]
“the unexpected result that the theory of spectral triples does not apply to the Carnot manifolds in the way one would expect. [p. 11] ”
“We will prove in this thesis that any horizontal Dirac operator on an arbitrary Carnot manifold cannot be hypoelliptic. This is a big difference to the classical case, where any Dirac operator is elliptic. [p. 12]”
It appears that the author reduces the problems to the Heisenberg groups. There is a solution, then, to use
R. Beals, P.C. Greiner, Calculus on Heisenberg manifolds, Princeton University Press, 1988
which gives something resembling spectral triples, but not quite all works, still:
“and show how hypoelliptic Heisenberg pseudodifferential operators furnishing a spectral triple and detecting in addition the Hausdorff dimension of the Heisenberg manifold can be constructed. We will suggest a few concrete operators, but it remains unclear whether one can detect or at least estimate the Carnot-Caratheodory metric from them. [p. 12]”
This seems to be an excellent article, more than that, because it is a phd dissertation many things are written clearly.
I am not surprised at all by this, it just means that, as in the case with the metric currents, there is an ingredient in the spectral triples theory which introduces by the backdoor some commutativity, which messes then with the non-commutative analysis (or calculus).
Instead I am even more convinced than ever that the minimal (!) description of sub-riemannian manifolds, as models of a non-commutative analysis, is given by dilation structures, explained most recently in arXiv:1206.3093 [math.MG].
A corollary of this is: sub-riemannian geometry (i.e. non-commutative analysis of dilation structures) is more non-commutative than non-commutative geometry .
I’m waiting for a negative result concerning the application of quantum groups to sub-riemannian geometry.