I am struggling for an exercice in differential geometry :
S is a minimal surface if and only if the Gauss map $N : S \rightarrow S^2$ satisfies for all $p \in S$ and all $\omega_1, \omega_2 \ T_p(S)$ : $$\langle dN_p(\omega_1),dN_p(\omega_2)\rangle = \lambda(p) \langle \omega_1, \omega_2 \rangle $$ where $\lambda(p)≠ 0$
I tried to use the fact that The mean curvature $H(p) = \frac{1}{2} \frac{eG + gE -2Ff}{EG -F^2}$. But I don't really see a link with this.
If someone can help me it would be great ! Thank you in advance