On the Hausdorff volume in sub-Riemannian geometry

TL;DR

This paper investigates the Radon-Nikodym derivative of the Hausdorff measure in sub-Riemannian manifolds, revealing its smoothness properties and relation to the Popp volume, with implications for intrinsic volume comparisons.

Contribution

It establishes the continuity and smoothness properties of the Radon-Nikodym derivative and clarifies its relation to the Popp volume, especially in higher dimensions.

Findings

The Radon-Nikodym derivative equals the volume of the nilpotent approximation's unit ball.

It is continuous in all cases, smooth up to dimension 4, and C^3 (or C^4 on curves) starting from dimension 5.

In higher dimensions, the derivative may not be proportional to the Popp volume due to dependence on the point.

Abstract

For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, while starting from dimension 5, in corank 1 case, it is C^3 (and C^4 on every smooth curve) but in general not C^5. These results answer to a question addressed by Montgomery about the relation between two intrinsic volumes that can be defined in a sub-Riemannian manifold, namely the Popp and the Hausdorff volume. If the nilpotent approximation depends on the point (that may happen starting from dimension 5), then they are not proportional, in general.