Given a smooth, rotationally symmetric Gaussian curvature profile, $G(r)$, how can we know if it is embeddable in $R^3$?
$G(r)$ is defined on $r=[0,R)$ where $r$ is the geodesic length from a fixed point and $R$ is finite. (So $G$ is defined on a disc).
Note that the Gaussian curvature is symmetric under rotations in the material coordinates, but the embedded shape need not be rotationally symmetric in $R^3$. For instance, a saddle with constant negative curvature has a rotationally symmetric $G(r)$ but is not a surface of revolution. In the image below, $G$ is constant along the green geodesic circles.
In other words, are there criteria for whether or not we can build a surface which has the given Gaussian curvature distribution?