Let's suppose we know the shape operator (aka Weingarten operator) of a given surface everywhere in its domain. Is there any way, analytical or numerical, to find the family of surfaces having the given shape operator?

And what if we know both first and second fundamental forms?

I know it's a very general problem. I will appreciate any clue.


Second part: If both fundamental forms are are known, then the surface is intrinsically fixed (up to translation and rotation in 3-space) as per Fundamental theorem of surface theory.

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