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 C 1 submanifolds in Carnot groups are generically transversal, the previous results prove that the “intrinsic measure” of C 1 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 C 1 transversal submanifolds.
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The first author acknowledges the support of the European Project ERC AdG *GeMeThNES*.
The second author acknowledges the support of the US National Science Foundation Grants DMS-0901620 and DMS-1201875.
The third author acknowledges the support of MIUR, GNAMPA of INDAM (Italy), University of Padova, Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: models, analysis, and control-theoretic problems” and University of Padova research project “Some analytic and differential geometric aspects in Nonlinear Control Theory, with applications to Mechanics”.
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Magnani, V., Tyson, J.T. & Vittone, D. On transversal submanifolds and their measure. JAMA 125, 319–351 (2015). https://doi.org/10.1007/s11854-015-0010-8
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DOI: https://doi.org/10.1007/s11854-015-0010-8