On transversal submanifolds and their measure
J. Anal. Math.
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 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-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, only depending on the embedding of the submanifold, that computes the Carnot-Carathéodory Hausdorff dimension of $C^1$ transversal submanifolds.
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