Metric Geometry of Carnot–Carathéodory Spaces and Its Applications

We describe new fine properties of Carnot–Carathéodory spaces under minimal assumptions on smoothness of the basis vector fields. As a corollary, we discover new geometric properties of weighted Carnot-Caratheodory spaces. All these results are new even for a “smooth” case. They play crucial role in the development of the differentiability theory on sub-Riemannian structures (see, e.g., works by S. Vodopyanov [1, 2]), in the investigation of non-equiregular Carnot–Carathéodory spaces (see, e.g., work by S. Selivanova [3]) and imply many basic results of the theory of non-holonomic spaces (see, e.g., work by S. Basalaev and S. Vodopyanov [4]).
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Definition (see, e.g., [5, 2, 4, 6, 7]). Fix a connected Riemannian $C^{\infty}$-manifold $\mathbb M$ of topological dimension $N$. The manifold $\mathbb M$ is called the Carnot–Carathéodory space if the tangent bundle $T\mathbb M$ has a filtration
$$ H\mathbb M=H_1\mathbb M\subsetneq\ldots\subsetneq H_i\mathbb M\subsetneq\ldots\subsetneq H_M\mathbb M=T\mathbb M $$
by subbundles such that every point $p\in\mathbb M$ has a neighborhood $U\subset\mathbb M$ equipped with a collection of $C^{1}$-smooth vector fields $X_1,\dots,X_N$ enjoying the following two properties.
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(1) At every point $v\in U$ we have a subspace
$$ H_i\mathbb M(v)=H_i(v)=\operatorname{span}\{X_1(v),\dots,X_{\dim H_i}(v)\}\subset T_v\mathbb M $$
of the dimension $\dim H_i$ independent of $v$, $i=1,\ldots,M$.
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(2) The inclusion $[H_i, H_j]\subset H_{i+j}$, $i+j\leq M$, holds.
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Moreover, if the third condition holds then the Carnot–Carathéodory space is called the Carnot manifold:
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(3) $H_{j+1}=\operatorname{span}\{H_j, [H_1,H_{j}], [H_2,H_{j-1}],\ldots,[H_k, H_{j+1-k}]\}$, where $k=\bigl[\frac{j+1}{2}\bigr]$, $H_0=\{0\}$, $j=1,\ldots, M-1$.
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The subbundle $H\mathbb M$ is called horizontal.
The number $M$ is called the depth of the manifold $\mathbb M$.
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The main result is the following
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Theorem [6, 7]. Let $\mathbb M$ be a Carnot–Carathéodory space with $C^{1,\alpha}$-smooth basis vector fields, $\alpha\geq0$ (if $\alpha=0$ then the fields belong to the class $C^1$). Then for each point of $\mathbb M$, there exists a sufficiently small neighborhood $\mathcal U\Subset\mathbb M$ possessing the following property: for $u,v\in \mathcal U$, $w=\gamma(1)$ and $\widehat{w}=\widehat{\gamma}(1)$, where $\gamma,\widehat{\gamma}:[0,1]\to\mathbb M$ are absolutely continuous (in the classical sense) curves contained in $\operatorname{Box}(u, \varepsilon)$ such that
\begin{equation*} \dot{\gamma}(t)=\sum\limits_{i=1}^Nb_i(t)X_i(\gamma(t)),\ \gamma(0)=v, \quad\text{and}\quad \dot{\widehat{\gamma}}(t)=\sum\limits_{i=1}^Nb_i(t)\widehat{X}^u_i(\gamma(t)),\ \widehat{\gamma}(0)=v, \end{equation*}
and each measurable function $b_i(t)$ meets the property
$$ \tag{1} \int_0^1|b_i(t)|\,dt<S\varepsilon^{\operatorname{deg}X_i}, $$
$S<\infty$, $i=1,\ldots, N$, we have
$$ \tag{2} \max\{d_{\infty}(w,\widehat{w}), d^u_{\infty}(w,\widehat{w})\}= \begin{cases} O(1)\cdot\varepsilon^{1+\frac{\alpha}{M}} \text{ if }\alpha>0,\\ o(1)\cdot\varepsilon\text{ if }\alpha=0, \end{cases} $$
with $O(1)$ and $o(1)$ to be uniform in $u\in\mathcal U$ and all collections of functions $\{b_i(t)\}_{i=1}^N$ with the property (1).
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Remark [7] (see also [8]). For weighted Carnot–Carathéodory spaces, the estimate in (2) is $O(1)\cdot\varepsilon^{1+\frac{\alpha}{l_M}}$ for $\alpha>0$, where $l_M$ is the maximal weight [3, 7, 8].
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The research is supported by Grant of the Government of Russian Federation for the State Support of Researches (Agreement No 14.B25.31.0029).