I bet there must be some research about this topic I'm not able to find. Suppose I have a smooth surface patch $\sigma_0$ I want to construct a local isometry $\sigma_{\infty}$ with some specific property, for example the curvature of $\sigma_\infty$ must be $0$ ideally (or in practice as low as possible).
Formally speaking I'm looking for a representation of a local isometry that flats. The first problem I struggle with is what class of surfaces $\sigma_0$ have the property that can be flattened. Assuming the input can be transformed into a flat one the only way I thought of constructing such transformation would be by means of a differential equation
$$ \frac{\partial \sigma}{\partial t} = f(D\sigma) $$
($D$ is a differential operator) or maybe using a second order differential equation. I have the feeling a problem like this in these terms has been already tackled ad I'd like to see some papers, but I really struggle to find anything.
Can anyone suggest some readings maybe?
I believe it can be modelled as
$$ \left\{ \begin{array}{l} \frac{d}{dt} \left( J_{\sigma_t}^T J_{\sigma_t}^T \right) = 0 \\ \sigma_0 = \text{given} \\ \mathcal{W}_{\sigma_{\infty}} = 0 \end{array} \right. $$
Where $\mathcal{W}$ is the shape operator (2 x 2 matrix describing the curvature).
Thank you