Opens of helicoid and catenoid Helicoid (H) and Catenoid (C) are locally isometrics. It means that there is a locally diffeomorphims $\phi: U \subset H \longrightarrow V\subset C$ such that $\left<d\phi_{p}(w_{1}), d\phi_{p}(w_{2})\right>_{\phi(p)} = \left<w_{1}, w_{2}\right>_{p}$ for $w_{1}, w_{2} \in T_{p} H$. But, if it is a diffeomorphism in $U$ to $V$, its inverse $\phi$ is continuous, then it leads opens into opens, but none open in helicoid is a open in the catenoid because they have differents curvature's lines. It is a paradox? Or am I making a mistake? Thanks for listening.

Whether two surfaces are locally isometric is an intrinsic question: There is a local diffeomorphism that preserves the _first_ fundamental form. This says nothing about the extrinsic geometry, i.e., the Gauss map or the second fundamental form. Thus, we expect no relation between the lines of curvature or the asymptotic curves of the two surfaces.