sHello to everyone. I have a problem with checking the Gauss-Codazzi compatibility equations by choosing the curvilinear coordinate system I'm going to list. Let E, F, G be the functions of the first fundamental form, and, f, g the functions of the second fundamental form, R1 and R2 are the main curvature rays and be A1 and A2 the Lamè's parameters. It is assumed to take a orthogonal and main curvilinear such that: $$ds^2=(R1)^2 (d\alpha_1)^2+(R2)^2 (d\alpha_2)^2$$ Where $$\alpha_1,\alpha_2$$ are the parametric coordinates of the surface which in this case coincide with the angular angular angles of the x and y axes $$r(\alpha_1,\alpha_2)=( x(\alpha_1,\alpha_2),y(\alpha_1,\alpha_2),z(\alpha_1,\alpha_2) )$$
this implies that $$F=f=0, A_1=R_1, A_2=R_2, E^{1/2}=R_1,g^{1/2}=R_2$$ Where it is important to specify that $$R_1>0,R_2>0, R_1<\infty,R_2<\infty$$
Recalling the Gauss-Codazzi's equations :
(1)$$\frac{d(\frac{A_1}{R_1})}{d\alpha_2}=\frac{1}{R_2}\frac{dA_1}{d\alpha_1}$$ (2)$$\frac{d(\frac{A_2}{R_2})}{d\alpha_1}=\frac{1}{R_1}\frac{dA_2}{d\alpha_1}$$ (3) $$\frac{d}{d\alpha_1}(\frac{1}{A_1} \frac{dA_2}{d\alpha1})+\frac{d}{d\alpha_2}(\frac{1}{A_2} \frac{dA_1}{d\alpha2})=-\frac{A_1A_2}{R_1R_2}$$
The (1) gives : $$\frac{dR_1}{d\alpha_2}=0$$ The (2) gives: :$$\frac{dR_2}{d\alpha_1}=0$$ But the (3) is non satisfied as it returns $$1=0$$
Same problem if instead I use the main intrinsic curvilinear coordinate system (analogous) $$ds^2=ds_1^2+ds_2^2$$ where in this case $$A_1=1,A_2=2$$
I get it from the Gauss-Codazzi 's equations (K1=1/R1 and K2=1/R2 are the mean curvature): $$(1)---> \frac{d(\frac{1}{R_1})}{ds_2}=\frac{dK_1}{ds_2}=0 $$ $$(2)---> \frac{d(\frac{1}{R_2})}{ds_1}=\frac{dK_2}{ds_1}=0 $$ The third (3) is satisfied $$\Leftrightarrow R_1R_2-->\infty$$ ie the surface is a plan.
Definitely i'm wrong about something conceptual but i can not understand what. I am obliged, for the problem I am working on, to take orthogonal and main curvilinear coordinates of that type. If anyone could make me understand where I'm wrong, it would make me a great favor, because the choice I've shown you is my only way. Thank you for your time.