I'm reading an article on singular Riemannian foliations and I have a few very basic questions:
(1) Apparently, the transnormallity condition is equivalent to say that the leaves are locally equidistant. However, to this moment, I can't even wrap my mind around the concept of distance between leaves. If a foliation is a partition of a manifold, how can there be a distance (greater than zero) between the leaves? I'm sorry if it's a stupid question, but I'm quite confused here.
(2) Also, I couldn't understand the idea of maximal dimension, so far. What does it mean a foliation have maximal dimension?
Thank you in advance if you can help.
"Locally equidistant" means that locally, if you take two leaves $L$ and $L'$ of the foliation, then every point in $L$ is the same distance from $L'$. A basic example to have in mind is the foliation of $\mathbb{R}^2$ whose leaves are the horizontal lines. This is locally equidistant--if you have two parallel lines, then the distance between a point on one line and the other line is independent of the point chosen on the first line.
In a singular foliation, the leaves have different dimensions. The "maximal dimension" then just refers to the largest dimension of any of the leaves of the foliation.