Carnot-Carathéodory空间的几何、可微性和余面积公式
Geometry of Carnot--Carathéodory Spaces, Differentiability and Coarea Formula
摘要：
本文给出了Gromov关于向量场幂零化定理的一个简单证明，并给出了一种新的方法，用于在具有$C^{1,α}$ -光滑基向量场的Carnot-Carathéodory空间中获得两个不同局部Carnot群的比较几何的定量估计，$α\in[0,1]$ .从这里，我们得到类似的估计比较几何的卡诺-Carathéodory空间和局部卡诺群。这两个定理包含了该理论的基本结果：Gromov型局部逼近定理，以及对于$α>0$ Rashevski的\v-Chow定理和Ball-Box定理等.我们应用所得结果证明了Carnot-Carathéodory空间中具有连续水平导数的映射的$hc$ -可微性.后者用于证明Carnot-Carathéodory空间中某些类切触映射的余面积公式。
英文摘要：
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, $α\in[0,1]$. 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.
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