Gaussian Curvature of the half-plane only with the metric

Question

Let $S= \{(u,v): u \gt 0 \}$ the half-plane. And the metric

$$g=du^2+u^2dv^2$$

How to find the Gaussian curvature of S? I don't if there is sufficiently information to do this.

Answer 1

Cool, so if you're using moving frames, start with $\omega_1=du$, $\omega_2=u\,dv$. Determine the connection form $\omega_{12}$ with $d\omega_1=\omega_{12}\wedge\omega_2$ and $d\omega_2=\omega_1\wedge\omega_{12}$. What do you get?