In my differential geometry course isometric surfaces were defined as having parametrizations $\textbf{r}_{1} = \textbf{f}(u,v)$ and $\textbf{r}_{2} = \textbf{f}(u,v)$ such that $\textbf{r}_{1}$, $\textbf{r}_{2}$ : $D \rightarrow{}\mathbb{R}^3$, where $D$ is open connected subset of $\mathbb{R}^2$, and for each plane curve $l$ on $D$ lengths of their images are equal: $|\textbf{r}_1(l)| = |\textbf{r}_2(l)|$.
To proof that the first fundamental form completely describes the metric properties of a surface we need to proof:
- That surfaces with equal coefficients of the first fundamental form are isometric
- That any isometric surfaces have parametrizations with equal coefficients of the first fundamental form.
Length of the curves on the surface are calcutated with coefficients of the first fundamental form, so it's quite easy to proof that equality of them implies surfaces' isometry.
But how can the second statement be proved?