It's straightforward to calculate the fundamental forms and the Gaussian curvature starting with a given parametrization. But how can I calculate the Gaussian curvature, given only the first fundamental form?
For example, if we have: $I= \dfrac{(du)^2+dv)^2}{[1+\frac{c}{4}(u^2+v^2)]^2}$, how should I proceed to get the Gaussian curvature?
Thanks for any advice!