If $I=(du)^2+\sin^2u(dv)^2=II$ then show that the surface is given by $$\underline{x}=\sin u\cos v\:e_1+\sin u\sin v\:e_2+\cos u\:e_3+\underline{c}$$
This exercise question is really interesting like the fundamental form are enough to determine the surface, but my book didn't show anything similar. Like I know here,
$$E=1,F=0,G=\sin^2 u$$ $$e=1,f=0,g=\sin^2 u$$ Now I stuck, like what approach should I follow to get the surface equation? Some random thought suggest to find curvature ($K=\frac{eg-f^2}{EG-F^2}=\frac{\sin^2 u}{\sin^2 u}=1$) or normal of surface which might help to determine the surface but I didn't get any equation like the question.
It seems using Weingarten equation and Gauss equation I got $X_{uuu}+X_u=0$ which actually led me to the surface equation (up to rigid motion: translation and rotation). Does this approach is general enough? I got another M.SE thread where it suggest to verify compatibility equations (Gauss' equation and the Codazzi-Mainardi Equations). Does it need so?