Showing to be Unit-sphere

Question

First and second fundamental forms are both $du^2 +\cos^2 u dv^2$

I want to show that the surface is a part of the unit sphere.

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What I did is following;

$E=L=1$

$F=M=0$

$N=G=\cos^2 u$

Verified all christoffel symbols

$\Gamma_{11}^1=\Gamma_{11}^2= \Gamma_{12}^1=\Gamma_{22}^2=0$

$\Gamma_{12}^2=-\tan u$

$\Gamma_{22}^1=\cos u\sin u$

I also find Gaussian curvature $K=1$.

$L=\sigma_{uu}\mathbf N$ $\Rightarrow$ $\sigma_{uu}=\mathbf N$

Similarly, $\sigma_{uv}=0$

By weingarden map $W=F_{I}^{-1}F_{II}=\left [\begin{matrix} E & F \\ F & G \end{matrix}\right ]^{-1}=\left [\begin{matrix} L & M \\ M & N \end{matrix}\right ]=\left [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right ]=\left [\begin{matrix} a& c \\ b & d \end{matrix}\right ]$

Then, $-\mathbf N_u=a\sigma_u +b\sigma_v$ $\Rightarrow$ $-\mathbf {N_u}=\sigma_u$

Similarly, $-\mathbf {N_v}=\sigma_v$

after there, please help me showing this question. Thank you.

Answer 1

Hint:

1. The surface is determined uniquely up to transformation using its first and second forms
2. If we get a parametrization of the unit sphere that gives these first and second fundamental forms, then these two surfaces coincides.
3. Consider the parametrization of the unit sphere as $$r(u,v)=(\cos u\cos v,\cos u\sin v, \sin u)$$
4. Find the first and second fundamental forms of the unit sphere and show that they are the same as the given surface.