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arXiv · 1211.6607

On transversal submanifolds and their measure

Valentino Magnani·Jeremy T. Tyson·Davide Vittone
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Abstract

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 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-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 C^1 transversal submanifolds.

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Valentino Magnani, Jeremy T. Tyson, Davide Vittone. 2012-11-28. On transversal submanifolds and their measure. https://arxiv.org/abs/1211.6607

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