A Coarea Formula for Smooth Contact Mappings of Carnot--Carathéodory Spaces

Details A Coarea Formula for Smooth Contact Mappings of Carnot--Carathéodory Spaces A Coarea Formula for Smooth Contact Mappings of Carnot--Carathéodory Spaces

2011-06-22

arxiv.org/abs/1106.4391v2

Mathematics

Metric Geometry

56 pages

Scientific paper

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.

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