English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

This paper is related to the problem of finding a good notion ofrectifiability in sub-Riemannian geometry. In particular, we study which kindof results can be expected for smooth hypersurfaces in Carnot groups. Our maincontribution will be a consequence of the following result: there exists a$C^{\infty}$ hypersurface $S$ without characteristic points that hasuncountably many pairwise non-isomorphic tangent groups on everypositive-measure subset. The example is found in a Carnot group of topologicaldimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorffmeasure $\mathcal{H}^{12}$. As a consequence, we show that for every Carnotgroup of Hausdorff dimension 12, any Lipschitz map defined on a subset of itwith values in $S$ has $\mathcal{H}^{12}$-null image. In particular, we deducethat this smooth hypersurface cannot be Lipschitz parametrizable by countablymany maps each defined on some subset of some Carnot group of Hausdorffdimension $12$. As main consequence we have that a notion of rectifiabilityproposed by S.Pauls is not equivalent to one proposed by B.Franchi, R.Serapioniand F.Serra Cassano, at least for arbitrary Carnot groups. In addition, we showthat, 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$\mathcal{H}^{12}(f(U))=0$. Finally, we prove that such an example does notexist in Heisenberg groups: we prove that all $C^{\infty}$-hypersurfaces in$\mathbb H^n$ with $n\geq 2$ are countably $\mathbb{H}^{n-1}\times\mathbbR$-rectifiabile according to Pauls' definition, even with bi-Lipschitz maps.

Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces