I'm not able to understand how the answer given above has been obtained. How did they deduce the sigma is isometric to the plane? Also, if a surface is isometric to another surface then does that mean they have the same first fundamental form and the same second fundamental form? Also, if a surface is isometric to another surface then is it true that they will have the same geodesics?
How else in general do we find the geodesics of a surface?
Any help would be much appreciated.
So let $S$ be the surface. There is a parametrization of $S$, let $\phi :\mathbb{R}^2\rightarrow \mathbb{R}^3$ with $\sigma$ being its restriction to $(0,1)\times (0,1)$. Note that the image of $\phi$ is $S$.
The surface $S$ has the same first fundamental form as the plane $\mathbb{R}^2$ which implies that they are locally isometric. Moreover $\sigma$ is differentiable. So we have a differentiable map that is a local isometry with $\mathbb{R}^2$ complete, connected riemannian manifold and $S$ connected riemannian manifold. This implies that they are isometric. See How to go from local to global isometry.
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Regarding the question about geodesics of isometric surfaces: If $\phi: M\rightarrow N$ is an isometry between two surfaces then the image of a geodesic in $M$, is a geodesic in $N$. They are not the same though.