Let the operator $L$ be given by $Lf(x):=\nabla\cdot (A\nabla f(x))$, where $f:\mathbb{R}^d\rightarrow \mathbb{R}$ belongs to a suitable class of functions $\mathcal{A}$. The carre du champ operator $\Gamma$ on $\mathcal{A}\times\mathcal{A}$ is then given by $\Gamma(f,f)(x)=(\nabla f(x))^TA(x)\nabla f(x)$. For suitable matrix $A$, $\Gamma(f,f)^{1/2}$ is nothing else then the length of the gradient $\nabla_Mf(x)$ on the Riemannian manifold $(\mathbb{R}^d,A^{-1})$, where $A^{-1}$ denotes the metric on $\mathbb{R}^d$ induced by $x\mapsto A(x)^{-1}$. On this manifold, the distance between two points is given by $d_M(x,y)=\inf_{\gamma\in\Omega(x,y)}\int_0^1\sqrt{\gamma'(t)^TA(\gamma(t))^{-1}\gamma'(t)}dt$, where the inf runs over all Lipschitz paths $\gamma$ such that $\gamma(0)=x$ and $\gamma(1)=y.$
The operator $L$ induces also another metric on $\mathbb{R}^d$. A curve $\gamma$ is called subunit for $L$ if $\vert \gamma'(t)\cdot\xi\vert^2\leq \xi^T A(\gamma(t))\xi$ for all $\xi\in\mathbb{R}^d$. Let $C_{sub}(x,y)$ denote the set of all subunit curves for $L$ that connect $x$ and $y$. We then write $d_L(x,y)=\inf_{\gamma\in C_{sub}(x,y)}\{b:\gamma(0)=x,\gamma(b)=y\}$. Note that the inf is taken over all subunit curves $\gamma:[0,b]\rightarrow \mathbb{R}^d$ which connect $x$ and $y$ at times $t=0$ and $t=b$ respectively. $d_L(x,y)$ is then the smallest such $b$.
Are the metrics $d_M$ and $d_L$ the same or at least equivalent?
I am not very versed in differential geometry so I might be missing something obvious. Bakry and Ledoux use the above ideas (first paragraph) to motivate the name carre du champ. They then use the intrinsic metric $d_L$ (albeit another definition equivalent to this one) associated to $L$ without making explicit any links betweens these ideas. Since this field is also called geometry of markov diffusion operators I am curious to find more motivation regarding the geometric content for these ideas.
