On transversal submanifolds and their measure

Valentino Magnani, Jeremy T. Tyson, Davide Vittone

Research output: Contribution to journalArticlepeer-review

Abstract

We study the class of transversal submanifolds in Carnot groups. We characterize their blow-ups at transversal points and prove a negligibility theorem for their “generalized characteristic set”, with respect to the Carnot-Carathéodory Hausdorff measure. This set is made up of all points of non-maximal degree. In light of the fact that C1 submanifolds in Carnot groups are generically transversal, the previous results prove that the “intrinsic measure” of C1 submanifolds is generically equivalent to their Carnot-Carathéodory Hausdorff measure. As a result, the restriction of this Hausdorff measure to the submanifold can be replaced by a more manageable integral formula that should be seen as a “sub-Riemannian mass”. Another consequence of these results is an explicit formula, depending only on the embedding of the submanifold, that computes the Carnot-Carathéodory Hausdorff dimension of C1 transversal submanifolds.

Original languageEnglish (US)
Pages (from-to)319-351
Number of pages33
JournalJournal d'Analyse Mathematique
Volume125
Issue number1
DOIs
StatePublished - Jan 2015

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)

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