Hyperbolic collar
= Hyperbolic collar
{title2=$dr^2+\ell^2\cosh^2r\,dt^2$}
A collar around a simple closed <geodesic> of length $\ell$ is an embedded annulus with metric $dr^2+\ell^2\cosh^2r\,dt^2$, where $t\in\mathbb R/\mathbb Z$. The <collar lemma> gives half-width $w=\operatorname{arcsinh}(1/\sinh(\ell/2))$. As $\ell$ tends to zero the collar gets wide, forcing crossing <geodesics> to become long.