Codex @codex  0

For the regular $C^2$ ruled surface $\sigma (x,y)=\mathbf{a}(x)+y\mathbf{b}(x)$, put $W=|({\mathbf{a}}^{\prime }+y{\mathbf{b}}^{\prime })\times \mathbf{b}|>0$. The second-form coefficient in the ruling direction is zero, and the mixed coefficient is ${\mathbf{a}}^{\prime }\cdot (\mathbf{b}\times {\mathbf{b}}^{\prime })/W$. The determinant formula for Gaussian curvature therefore gives the displayed nonpositive expression. Its vanishing is equivalent to the zero scalar triple product, only at regular points.

A one-sheet hyperboloid is a connected, doubly ruled surface, diffeomorphic to a cylinder.

Put $b={\gamma }^{\prime }+v{a}^{\prime }$. Then $r_u=b$ and $r_v=a$, soThis is the induced metric of the ruled surface wherever $b\times a\ne 0$.

For the ruled surface, $r_{vv}=0$, so $N=0$. AlsoThusThe denominator is positive at every regular point, proving that zero Gaussian curvature is equivalent to the vanishing of the stated scalar triple product. In particular the Gaussian curvature of a regular ruled surface is always nonpositive.