Let $\mathcal{F}$ be a codimension-1 continuous foliation of $\mathbb{R}^2$ with $C^1$ leaves. That is, a partition of $\mathbb{R}^2$ into $C^1$ curves which can locally be mapped continuously to parallel lines. Define a distance along a leaf $d_{\mathcal{F}}: \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}$ by: If $x$ and $y$ lie on the same leaf of the foliation, then $d_{\mathcal{F}}(x,y)$ is the length of the leaf segment between them, and if they lie on different leaves, set $d_{\mathcal{F}}(x,y)=\infty$.
I want to show:
If the leaves of $\mathcal{F}$ are tangent to a uniformly continuous vector field $X$ on $\mathbb{R}^2$, then the map $d_{\mathcal{F}}(x,y)$ is uniformly continuous with respect to the standard metric on $\mathbb{R}^2\times\mathbb{R}^2$.
This arises in articles and books I have read about dynamical systems, often in more complicated contexts, but I'm using this setting at a starting point. However, I do not have a background in foliations, and I have just been trying to use basic differential geometry to show this.
It suffices to show that given $\varepsilon>0$, there is $\delta>0$ so that $\|x-y\| <\delta \implies \|d_{\mathcal{F}}(x,y)\|<\varepsilon$. However I struggle to proceed: I first try and show the result for $d_{\mathcal{F}}$ restricted to a leaf, where I use a parametrisation of this leaf and use the definition for distance of a curve. However, I cannot manage to relate this to the property that the leaf is tangent to the vector fied $X$. If I assume the result holds when restricted to a leaf, I expect to use a foliation chart to extend the result to $\mathbb{R}^2$, however I am also unsure of how this argument would follow.