When does the shape operator commute with a derivative?

Question

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.

Answer 1

The question is essentially equivalent to the following classical question: Given a smooth surface $S$ and a bundle map $L:TS\to TS$, when does there exist an immersion $x:S\to \mathbb{R}^3$ such that $L$ is the induced shape operator of $x$ (and, when one such $x$ does exist, how many essentially distinct such $x$ exist)? [Two immersions $x, y:S\to\mathbb{R}^3$ are essentially distinct if $x$ cannot be obtained from $y$ by composing with a rigid motion.]

I wrote an article about this problem, On surfaces with prescribed shape operator, Results in Mathematics 40 (2001), 88–121, that describes what was known about it in 2001 and gives references to the literature (which is considerable). Unfortunately, the journal inadvertently published my original submission rather than the revised version that I prepared after the referee's report, so it would actually be better to consult the version posted on the arXiv: arXiv:math/0107083. (The editors of RiM published a note in the journal about the publication mishap. Also, see the Math Reviews entry MR1898190 for more information.)

Here is the abstract for my article:

The problem of immersing a simply connected surface with a prescribed shape operator is discussed. From classical and more recent work, it is known that, aside from some special degenerate cases, such as when the shape operator can be realized by a surface with one family of principal curves being geodesic, the 'space' of essentially distinct realizations is a convex set in an affine space of dimension at most 3. The cases where this maximum dimension of realizability is achieved have been classified, and it is known that there are two such families of shape operators, one depending essentially on three arbitrary functions of one variable (called Type I in this article) and another depending essentially on two arbitrary functions of one variable (called Type II in this article). In this article, these classification results are rederived, with an emphasis on explicit computability of the space of solutions. It is shown that, for operators of either type, their realizations by immersions can be computed by quadrature. Moreover, explicit normal forms for each can be computed by quadrature together with, in the case of Type I, by solving a single linear second order ODE in one variable. (Even this last step can be avoided in most Type I cases.) The space of realizations is discussed in each case, along with some of their remarkable geometric properties. Several explicit examples are constructed (mostly already in the literature) and used to illustrate various features of the problem.