Lecture 17. The Carnot Carathéodory distance

In this Lecture we introduce a canonical distance on a Carnot group. This distance is naturally associated to the sub-Riemannian structure which is carried by a Carnot group. It plays a fundamental role in the rough paths topology. Let $\mathbb{G}_N(\mathbb{R}^d)$ be the free Carnot group over $\mathbb{R}^d$ . Remember that if $x \in C^{1-var}([0,T],\mathbb{R}^d)$ , then we denote by $S_N(x)$ the lift of $x$ in $\mathbb{G}_N(\mathbb{R}^d)$ . The first important concept is the notion of horizontal curve.

Definition: A curve $y: [0,1] \to \mathbb{G}_N(\mathbb{R}^d)$ is said to be horizontal if there exists $x \in C^{1-var}([0,T],\mathbb{R}^d)$ such that $y=S_N(x)$ .

It is remarkable that any two points of $\mathbb{G}_N(\mathbb{R}^d)$ can be connected by a horizontal curve.

Proposition: Given two points $g_1$ and $g_2 \in \mathbb{G}_N(\mathbb{R}^d)$ , there is at least one $x \in C^{1-var}([0,T],\mathbb{R}^d)$ such that $g_1S_N(x)(1)=g_2$ .

Proof: Let us denote by $G$ the subgroup of diffeomorphisms $\mathbb{G}_N(\mathbb{R}^d) \rightarrow \mathbb{G}_N(\mathbb{R}^d)$ generated by the one-parameter subgroups corresponding to $X_1, \cdots , X_d$ . The Lie algebra of $G$ can be identified with the Lie algebra generated by $X_1, \cdots , X_d$ , i.e. $\mathfrak{g}_N(\mathbb{R}^d)$ . We deduce that $G$ can be identified with $\mathbb{G}_N(\mathbb{R}^d)$ itself, so that it acts transitively on $\mathbb{G}$ . It means that for every $x \in \mathbb{G}_N(\mathbb{R}^d)$ , the map $G \rightarrow \mathbb{G}_N(\mathbb{R}^d)$ , $g \rightarrow g(x)$ is surjective. Thus, every two points in $\mathbb{G}$ can be joined by a piecewise smooth horizontal curve where each piece is a segment of an integral curve of one of the vector fields $\mathbf{e}_i$ $\square$

In the above proof, the horizontal curve constructed to join the two points is not smooth. Nevertheless, it can be shown that it is always possible to connect two points with a smooth horizontal curve.

Let us also remark that this theorem is a actually a very special case of the so-called Chow-Rashevski theorem which is one of the cornerstones of sub-Riemannian geometry. We now are ready for the definition of the Carnot-Carathéodory distance.

Definition For $g_1,g_2 \in \mathbb{G}_N(\mathbb{R}^d)$ , we define
$d(g_1,g_2)=\inf_{\mathcal{S}(g_1,g_2)} \| x \|_{1-var,[0,1]},$
where
$\mathcal{S}(g_1,g_2)=\{ x \in C^{1-var}([0,1],\mathbb{R}^d), g_1S_N(x)(1)=g_2 \}.$
$d(g_1,g_2)$ is called the Carnot-Carathéodory distance between $g_1$ and $g_2$ .

The first thing to prove is that $d$ is indeed a distance.
Lemma: The Carnot-Carathéodory distance is indeed a distance.

Proof: The symmetry and the triangle inequality are easy to check and we let the reader find the arguments. The last thing to prove is that $d(g_1,g_2)=0$ implies $g_1=g_2$ . From the definition of $d$ it clear that $d_R \le d$ where $d_R$ is the Riemmanian measure on $\mathbb{G}_N(\mathbb{R}^d)$ . It follows that $d(g_1,g_2)=0$ implies $g_1=g_2$ $\square$

We then observe the following properties of $d$ :

Proposition:

For $g_1,g_2 \in \mathbb{G}_N(\mathbb{R}^d)$ , $d(g_1,g_2)=d(g_2,g_1)=d(0,g_1^{-1} g_2).$

Let $(\Delta_t)_{t \ge 0}$ be the one parameter family of dilations on $\mathbb{G}_N(\mathbb{R}^d)$ . For $g_1,g_2 \in \mathbb{G}_N(\mathbb{R}^d)$ , and $t \ge 0$ , $d(\Delta_t g_1,\Delta_t g_2)=t d(g_1,g_2)$ .

Proof: The first part of the proposition stems from the fact that for every $x \in C^{1-var}([0,T],\mathbb{R}^d)$ , $S_N(x)^{-1}= S_N(-x)$ , so that $g_1S_N(x)(1)=g_2$ is equivalent to $g_2S_N(-x)(1)=g_1$ which also equivalent to $S_N(x)(1)=g_1^{-1} g_2$ . For the second part, we observe that for $t \ge 0$ , $\Delta_t S_N(x)=S_N(tx)$ $\square$

The Carnot-Carathéodory distance is pretty difficult to explicitly compute in general. It is often much more convenient to estimate using a so-called homogeneous norm.

Definition: A homogeneous norm on $\mathbb{G}_N(\mathbb{R}^d)$ is a continuous function $\parallel \cdot \parallel : \mathbb{G}_N(\mathbb{R}^d) \rightarrow [0,+\infty)$ , such that:

$\parallel \Delta_t x \parallel=t \parallel x \parallel$ , $t \ge 0$ , $x \in \mathbb{G}_N(\mathbb{R}^d)$ ;

$\parallel x^{-1} \parallel= \parallel x \parallel$ , $x \in \mathbb{G}_N(\mathbb{R}^d)$ ;

$\parallel x \parallel=0$ if and only if $x=0$ .

It turns out that the Carnot-Carathéodory distance is equivalent to any homogeneous norm in the following sense:

Theorem: Let $\parallel \cdot \parallel$ be a homogeneous norm on $\mathbb{G}_N(\mathbb{R}^d)$ . There exist two positive constants $C_1$ and $C_2$ such that for every $x,y \in \mathbb{G}_N(\mathbb{R}^d)$ ,
$A \| x^{-1}y \| \le d(x,y) \le B \| x^{-1}y \|.$

By using the left invariance of $d$ , it is of course enough to prove that for every $x \in \mathbb{G}_N(\mathbb{R}^d)$ ,
$A \| x \| \le d(0,x) \le B \| x \|.$
We first prove that the function $x\to d(0,x)$ is bounded on compact sets (of the Riemannian topology of the Lie group $\mathbb{G}_N(\mathbb{R}^d)$ ). As we have seen before, every $x \in \mathbb{G}_N(\mathbb{R}^d)$ can be written as a product:
$x=\prod_{i=1}^N e^{t_i X_{k_i}}.$
From the very definition of the distance, we have then
$d(0,x)\le d\left(0,\prod_{i=1}^N e^{t_i X_{k_i}}\right)\le \sum_{i=1}^N |t_i|.$
It is not difficult to see that $\sum_{i=1}^N |t_i|$ can uniformly be bounded on compact sets, therefore $d(0,x)$ is bounded on compact sets. Consider now the compact set
$\mathbf{K}= \{ x \in \mathbb{G}_N(\mathbb{R}^d), \| x \|=1 \}.$
Since $d(0,x)$ is bounded on $K$ , we deduce that there exists a constant $B$ such that for every $x \in \mathbf{K}$ ,
$d(0,x) \le B.$
Since $d_R \le d$ , where $d_R$ is the Riemannian distance, we deduce that there exists a constant $A$ such that for every $x \in \mathbf{K}$ ,
$d(0,x) \ge A.$
Now, for every $x \in \mathbb{G}_N(\mathbb{R}^d)$ , $x \neq 0$ , we deduce that
$A \le d\left( 0, \Delta_{1/\| x \|} x \right) \le B$
This yields the expected result $\square$

Let us give an example of a homogeneous norm which is particularly adapted to rough paths theory. Write the stratification of $\mathfrak{g}_N(\mathbb{R}^d)$ as:
$\mathfrak{g}_N(\mathbb{R}^d)=\mathcal{V}_1 \oplus \cdots \oplus \mathcal{V}_N$ and denote by $\pi_i$ the projection onto $\mathcal{V}_i$ . Let us denote by $\| \cdot \|$ the norm on $\mathfrak{g}_N(\mathbb{R}^d)$ that comes from the norm on formal series. Then, it is easily checked that
$\rho(g)=\sum_{i=1}^N \| \pi_i (g) \|^{1/i}$
is an homogeneous norm on $\mathbb{G}_N(\mathbb{R}^d)$ . This homogeneous norm is particulary adapted to the study of paths because if $x \in C^{1-var}([0,T], \mathbb{R}^d)$ , then one has:
$\rho\left( (S_N(x)(s))^{-1} S_N(x)(t) \right)=\sum_{k=1}^N \left\| \int_{\Delta^k [s,t]} dx^{\otimes k} \right\|^{1/k}$

We finally quote the following result, not difficult to prove which is often referred to as the ball-box estimate.

Proposition: There exists a constant $C$ such that for every $x,y \in \mathbb{G}_N(\mathbb{R}^d)$ ,
$d(x,y) \le C \max \{ \| x-y \|, \| x -y \|^{1/N} \max \{ 1, d(0,x)^{1-1/N} \} \}.$
and
$\| x-y \| \le C \max \{ d(x,y) \max \{ 1, d(0,x)^{N-1} \}, d(x,y)^N \}.$
In particular, for every compact set $K \subset \mathbb{G}_N(\mathbb{R}^d)$ , there is a constant $C_K$ such that for every $x,y \in K$ ,
$\frac{1}{C_K} \| x-y\| \le d(x,y) \le C_K \| x -y \|^{1/N}.$