A Coarea Formula for Smooth Contact Mappings of Carnot--Carath\'{e}odory Spaces

TL;DR

This paper establishes a coarea formula for smooth contact mappings in Carnot--Carathéodory spaces, analyzing level surfaces and measure comparisons, with implications for characteristic points and measure degeneracy.

Contribution

It introduces a coarea formula for contact mappings in Carnot manifolds and compares Riemannian and sub-Riemannian measures on level surfaces.

Findings

Sub-Riemannian measure of characteristic points is zero on almost every level set.

Sharp asymptotic behavior of measures is characterized.

Level surfaces' measure properties are clarified in Carnot--Carathéodory spaces.

Abstract

We prove the coarea formula for sufficiently smooth contact mappings of Carnot manifolds. In particular, we investigate level surfaces of these mappings, and compare Riemannian and sub-Riemannian measures on them. Our main tool is the sharp asymptotic behavior of the Riemannian measure of the intersection of a tangent plane to a level surface and a sub-Riemannian ball. This calculation in particular implies that the sub-Riemannian measure of the set of characteristic points (i.\,e., the points at which the sub-Riemannian differential is degenerate) equals zero on almost every level set.