There is this well-known Weyl Embedding theorem which guarantees that a metric of positive Gaussian curvature can be realized as a convex surface in $R^3$, which is unique up to rigid motion in $R^3$. The uniqueness then implies that the second fundamental form of the isometrically embedded surface is determined solely by the metric. But is there anywhere such a formula for the second fundamental form is written? Thanks in advance!
(Side mark: I already that Barnik's construction of prescribed scalar curvature could be used for scalar flat case but is there anything more general?)