Suppose we have a smooth map $\varphi : S\to H$ between two regular parametric surfaces in $\mathbb{R}^3.$ Then at any point $p\in S,$ we have following maps between corresponding tangent spaces: $\require{AMScd}$\begin{CD} T_pS @>d\varphi_p>> T_{\varphi(p)}H\\ @V dn_{S, p} V V @VV dn_{H, \varphi(p)} V\\ T_pS @>>d\varphi_p> T_{\varphi(p)}H \end{CD}
where $dn_{S, p}: T_pS\to T_pS$ is the shape operator (differential of the Gauss map). Now, I wonder about the necessary and sufficient conditions that make this diagram commutative. I have no idea, to begin with. Also, if any, what are the interesting geometric implications when these two linear maps commute?