The fundamental theorem of surface theory asserts that the existence of a simply connected surface in $\mathbb{R}^3$ whose first and second fundamental forms are $I$ and $II$, if the coefficients ($g_{ij}$) and ($h_{ij}$) of $I$ and $II$ satisfy the Gauss-Codazzi system.
Now if we only consider a given symmetric positive definite $2\times2$ matrix $g=(g_{ij})$, can we find compatible conditions only on $g$, such that $g$ is the first fundamental form of a simply connected surface in $\mathbb{R}^3$? Say, we need the existence of parametrization $y:\omega\subset\mathbb{R}^2\to\mathbb{R}^3$ such that $\nabla y^T\nabla y=g$.