how to construct radical function around a cusp?

Question

I read this construction from the paper "Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one": Let M be a rank 1 locally symmetric space. On each cusp $\hat{M}$ of $M$ there exists a proper function $r: \hat{M} \to R_+$ such that r is smooth with $|\nabla r|=1$ and r has compact level set. The metric g on $\hat{M}$ may then be written as $dr^2+ ^r g$ where $^rg$ is a metric on $\Sigma_0=r^{-1}(0)$. It seems to me that this function r works as the radical function as in the spherical coordinate. So it should be the distance to the cusp. Am I correct? Also, I do not understand that $^rg$ is a metric on $\Sigma_0=r^{-1}(0)$. If it is analogous to the spherical coordinate, it should look like $dr^2+ r\cdot g$ where g is the metric on $r^{-1}(1)$. I want to make sure if the original expression is a typo or I do not understand this correctly.

Answer 1

Take $N \subset M$ an orientable hypersurface and $\nu$ a normal field along $N$. Then the normal exponential map \begin{align} E : (-\varepsilon,\varepsilon) \times N &\longrightarrow M \\ (t,x) &\longmapsto \exp_x (t\nu_x) \end{align} is -under some curvature asumptions- a (local) diffeomorphism. Read on the left, the metrics $g$ of $M$ is \begin{align} E^*g = \mathrm{d}t^2 + g_t \end{align} with $(g_t)$ the restriction of $E^*g$ on $\{t\}\times N\simeq N$. You can think of $(g_t)$ as a family of metrics over $N$ with one parameter.

In this example, $r(x) = d(N,x)$ has $|\nabla r|=1$ and $r^{-1}(0) = N$.

In that view, it is similar to polar coordinates, where $N$ plays the role of the sphere.

To match more to your question: to construct such a function $r$, it seems you can prove that there exists a compact orientable hypersurface $N$ near a cusp for which the function $E$ is defined on $[0,+\infty)\times N$ and is a diffeomorphism onto its image. For example, in dimension $2$, take a (convec) non-contractible loop around the cusp.