Theorema egregium

Definition

$K=\frac{\det (\mathrm{I}\mathrm{I})}{\det (\mathrm{I})}=f(E,F,G)$ Here, the First Fundamental Form $\mathrm{I}=\left(\begin{array}{cc}E& F\\ F& G\end{array}\right)$ and the Second Fundamental Form $\mathrm{I}\mathrm{I}=\left(\begin{array}{cc}L& M\\ M& N\end{array}\right)$

Gaussian Curvature (sectional curvature) is completely determined by its Riemannian metric