Calculus of Variations and Geometric Measure Theory

V. Magnani - J. T. Tyson - D. Vittone

On transversal submanifolds and their measure

created by vittone on 28 Nov 2012
modified on 30 Jul 2020


Published Paper

Inserted: 28 nov 2012
Last Updated: 30 jul 2020

Journal: J. Anal. Math.
Volume: 125
Pages: 319–351
Year: 2015

ArXiv: 1211.6607 PDF


We study the class of transversal submanifolds. We characterize their blow-ups at transversal points and prove a negligibility theorem for their "generalized characteristic set", with respect to the Carnot-Carath\'eodory Hausdorff measure. This set is made by all points of non-maximal degree. Observing 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-Carath\'eodory 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, only depending on the embedding of the submanifold, that computes the Carnot-Carath\'eodory Hausdorff dimension of C1 transversal submanifolds.

Tags: GeMeThNES
Keywords: stratified groups, Submanifolds, Hausdorff measure