For an oriented surface in $\mathbb{R}^3$ the curvature tensor can be expressed as $$ R(X,Y)Z = \langle L(Y),Z \rangle L(X) - \langle L(X),Z \rangle L(Y) $$
where $X,Y,Z$ are vectors fields on the surface and $L$ is the shape operator.
I was wondering whether an inverse relationship exists, that allows to write the shape operator $L$ as a function of the curvature tensor.
I am a little confused in this regard, as I know that the curvature tensor is an intrinsic property of the surface, ultimately deriving from its metric, whereas the shape operator describes how the surface is embedded in $\mathbb{R}^3$.
In other words, if such a relationship existed, that allows the shape operator to be reconstructed from the curvature tensor (so from the metric) an ant living on a cilinder, that can calculate the curvature tensor (and, of course, the gaussian curvature) on each point, could also calculate the shape operator and ultimately reconstruct the embedding of his 2 dimensional world in $\mathbb{R}^3$.
Any help? thanks