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