What is a "simple" (local) isometry $F: P \rightarrow P_{+}$ where $$ P = \{(x,y,0) : x,y \in \mathbb{R}\} $$ and $$ P_{+} \{(x,y,0) : x,y \in \mathbb{R} \mbox{ with } y>0\}. $$ Here, local isometry $F: M \rightarrow N$ of surfaces is a mapping that preserves dot products of tangent vectors.
I have tried with maps by involucring functions like the exponential function, but have not been successful in achieving a such isometry or local isometry. In fact, I am trying to solve the following exercise:
Find two surfaces $S_1$ and $S_2$ such that $S_1$ is locally isometric to $S_ 2$ but $S_2$ is not locally isometric to $S_1$
Thanks in advance by any help