Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces

Abstract

This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in Carnot groups. Our main contribution will be a consequence of the following result: there exists a -hypersurface without characteristic points that has uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. The example is found in a Carnot group of topological dimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorff measure . As a consequence, we show that any Lipschitz map defined on a subset of a Carnot group of Hausdorff dimension 12, with values in , has negligible image with respect to the Hausdorff measure . In particular, we deduce that cannot be Lipschitz parametrizable by countably many maps each defined on some subset of some Carnot group of Hausdorff dimension 12. As main consequence we have that a notion of rectifiability proposed by S. Pauls is not equivalent to one proposed by B. Franchi, R. Serapioni and F. Serra Cassano, at least for arbitrary Carnot groups. In addition, we show that, given a subset of a homogeneous subgroup of Hausdorff dimension 12 of a Carnot group, every bi-Lipschitz map satisfies . Finally, we prove that such an example does not exist in Heisenberg groups: we prove that all -hypersurfaces in with are countably -rectifiable according to Pauls’ definition, even with bi-Lipschitz maps.