I'm dealing with a problem associated with helicoid-and-catenoid isometry. Paramatrize helicid by $f_{1}=(u\cos v,u\sin v,v)$, catenoid by $f_{2}=(\cosh z \cos\theta,\cosh z\sin\theta,z)$. Let $u=\sinh z$ and $v=\theta $, their first fundamental form become the same \begin{pmatrix} \cosh^{2}z &0 \\ 0 & \cosh^{2} \end{pmatrix}
However, I don't understand what's "this isometry is not the restriction of an isometry of $\mathbb{R}^{3}$" mean and don't know how to show this.