= Solution
Two regular coordinate patches are locally isometric under a coordinate correspondence exactly when it preserves their <first fundamental forms>. More explicitly, a <diffeomorphism> $\Phi:U\to\widetilde U$ of parameter domains induces a <local isometry> if and only if
$$
\boxed{\Phi^*\widetilde I=I.}
$$
In common coordinates this means equality of the three coefficient functions $E,F,G$. The criterion follows because the <first fundamental form> gives the <inner product> of every pair of <tangent vectors>. Equality therefore preserves lengths of all <curves>; conversely a smooth <isometry> preserves tangent-vector lengths and their <inner products> by polarization.
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