Given first fundamental form, provide an example of its surface

Question

How to find surface of this first fundamental form (not a plane)? I know it may sound stupid, but my few attempts got me $(u,v,0)$, but it is plane. Is there any algorithm, I could not find any.

Answer 1

If we want a surface embedded in $\mathbb{R}^3$, we can write a partial differential equation to find such surfaces.

Let $x: U\to\mathbb{R}^3$ be a patch with $ds^2=\lambda(du^2+dv^2)$. We can write this out in components as $$x(u,v)=(f(u,v),g(u,v),h(u,v))$$ with $f,g,h$ smooth functions from $\mathbb{R}^2\to\mathbb{R}$. Then $$x_u=(f_u,g_u,h_u) \text{ and } x_v=(f_v,g_v,h_v)$$ Our coordinate functions must satisfy $x_u\cdot x_u=\lambda$, $x_v\cdot x_u=0$ and $x_v\cdot x_v=\lambda$ giving us the following set of partial differential equations. \begin{align} x_u\cdot x_u&= f_u^2+g_u^2+h_u^2=\lambda\\ x_v\cdot x_u&=f_uf_v+g_ug_v+h_uh_v=0\\ x_v\cdot x_v&= f_v^2+g_v^2+h_v^2=\lambda \end{align} Solving these equations will give us a surface with the desired property.