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 (the derivative of the Gauss map). Now, I wonder about the necessary and sufficient conditions that make this diagram commutative. Also, if any, what are the interesting geometric implications when these two linear maps commute?

The exact same question was posted here on MathStackExchange 10 days ago but wasn't able to attract any attention.

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 (the derivative of the Gauss map). Now, I wonder about the necessary and sufficient conditions that make this diagram commutative. Also, if any, what are the interesting geometric implications when these two linear maps commute?

The exact same question was posted here on MathStackExchange 10 days ago but wasn't able to attract enough attention.