On transversal submanifolds and their measure

TL;DR

This paper investigates transversal submanifolds in Carnot groups, characterizing their blow-ups, proving measure negligibility results, and establishing formulas for intrinsic measures and Hausdorff dimensions, advancing understanding of sub-Riemannian geometry.

Contribution

It provides new characterizations of transversal submanifolds, measure equivalence results, and explicit formulas for their Hausdorff dimension in Carnot groups.

Findings

Intrinsic measure is generically equivalent to Carnot-Carathéodory Hausdorff measure for C^1 submanifolds.

Negligibility of the generalized characteristic set with respect to Hausdorff measure.

Explicit formula for the Hausdorff dimension based on embedding.

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.