The surface is parametrized by two variables $\sigma_1$ and $\sigma_2$. Moreover, this surface evolves in time. As a result, coordinates of the surface are: $\vec{F} =[x(\sigma_1,\sigma_2 , t), y(\sigma_1, \sigma_2, t), z(\sigma_1, \sigma_2, t)]$. Is it possible to chose such parametrization, that $∂_t\vec{F} · ∂_{\sigma_1} \vec{F} = ∂_t \vec{F} · ∂_{\sigma_2} \vec{F} = ∂_{\sigma_2} \vec{F} · ∂_{\sigma_1} \vec{F} = 0$ (set of orthogonal vectors), or: $$ ∂_t x ∂_{\sigma_1} x + ∂_ty ∂_{\sigma_1} y + ∂_t z ∂_{\sigma_1} z = 0;\\ ∂_t x ∂_{\sigma_2} x + ∂_ty ∂_{\sigma_2} y + ∂_t z ∂_{\sigma_2} z = 0;\\ ∂_{\sigma_1} x ∂_{\sigma_2} x + ∂_{\sigma_1} y ∂_{\sigma_2} y + ∂_{\sigma_1} z ∂_{\sigma_2} z = 0. $$ We a priori think that functions unbounded and smooth enough (at least $\in C^1$). $\sigma_1,\sigma_2\in\mathbb{R},t\in\mathbb{R_+}$.
I know FA a bit, so it's a question of well-posedness, I suppose. But I couldn't find any useful books or publications to help me. If you have any questions or you would like to have more information, I'm willing to provide it at any time.