Geometry of Carnot--Carathéodory Spaces, Differentiability and Coarea Formula

2008/04/21 by Karmanova, Maria, Vodopyanov, Sergey
#28A75 #53C17 #58C35 #93B29 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.0804.3291

Abstract

We give a simple proof of Gromov's Theorem on nilpotentization of vector fields, and exhibit a new method for obtaining quantitative estimates of comparing geometries of two different local Carnot groups in Carnot--Carathéodory spaces with C^1,α-smooth basis vector fields, α∈[0,1]. From here we obtain the similar estimates for comparing geometries of a Carnot--Carathéodory space and a local Carnot group. These two theorems imply basic results of the theory: Gromov type Local Approximation Theorems, and for α>0 Rashevski\vı-Chow Theorem and Ball--Box Theorem, etc. We apply the obtained results for proving hc-differentiability of mappings of Carnot--Carathéodory spaces with continuous horizontal derivatives. The latter is used in proving the coarea formula for some classes of contact mappings of Carnot--Carathéodory spaces.

Related

• Privileged coordinates for Carnot-Carathéodory spaces of low smoothness
• Metric geometry of nonregular weighted Carnot-Carathéodory spaces
• Submanifolds and differential forms on Carnot manifolds, after M. Gromov and M. Rumin
• A Coarea Formula for Smooth Contact Mappings of Carnot--Carathéodory Spaces
• Regularity estimates for convex functions in Carnot-Carathéodory spaces