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 $C^{\infty }$-hypersurface $S$ 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 ${\mathcal{H}}^{12}$. As a consequence, we show that any Lipschitz map defined on a subset of a Carnot group of Hausdorff dimension 12, with values in $S$, has negligible image with respect to the Hausdorff measure ${\mathcal{H}}^{12}$. In particular, we deduce that $S$ 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 $U$ of a homogeneous subgroup of Hausdorff dimension 12 of a Carnot group, every bi-Lipschitz map $f:U\to S$ satisfies ${\mathcal{H}}^{12}\left(f\left(U\right)\right)=0$. Finally, we prove that such an example does not exist in Heisenberg groups: we prove that all $C^{\infty }$-hypersurfaces in ${\mathbb{H}}^n$ with $n\ge 2$ are countably ${\mathbb{H}}^{n-1}\times \mathbb{R}$-rectifiable according to Pauls’ definition, even with bi-Lipschitz maps.

本文与在次黎曼几何中寻找可纠正性的良好概念有关。特别是，我们研究了在卡诺（Carnot）组中对光滑的超曲面可预期的结果。我们的主要贡献将是以下结果的结果：存在一个$C^{\infty }$-超表面 $\mathrm{小号}$没有特征点，每个特征量子集上都有无数成对的成对切线组。该示例位于拓扑维数为8的Carnot组中，其维数为Hausdorff，维数为12，因此我们在其上使用Hausdorff测度${\mathcal{H}}^{12}$。结果，我们证明了在Hausdorff维度12的Carnot组的子集上定义的任何Lipschitz映射，其值在$\mathrm{小号}$，关于Hausdorff测度的图像可忽略不计 ${\mathcal{H}}^{12}$。特别是，我们推断$\mathrm{小号}$不能通过在Hausdorff维度12的某些卡诺组的某个子集上定义的大量映射来对Lipschitz进行参数化。作为主要结果，我们认为S. Pauls提出的可纠正性概念不等于B. Franchi，R.提出的可纠正性概念。 Serapioni和F. Serra Cassano，至少适用于任意卡诺组合。此外，我们证明，给定一个子集$ü$ 每个bi-Lipschitz映射的Carnot组的Hausdorff维度12的齐次子组 $F：ü\to \mathrm{小号}$ 满足 ${\mathcal{H}}^{12}（F（ü））=0$。最后，我们证明在海森堡小组中不存在这样的例子：我们证明所有$C^{\infty }$-超曲面 ${\mathbb{H}}^{ñ}$ 与 $ñ\ge 2$ 数不胜数 ${\mathbb{H}}^{ñ-\mathrm{1个}}\times \mathbb{[R}$-可以根据Pauls的定义进行纠正，即使使用双Lipschitz映射也是如此。