Finding lines and curves on a surface

Question

Suppose I have some equation of surface $S$ (let's take $f(u,v) = (uv,u+v,u^2 + v^2)$ for example). How do I find if there are some lines on a surface? If there are some lines on this surface, how do I understand if this is a ruled surface?

Answer 1

As Lee and John suggested in comments, in order for a line through $P\in S$ to lie entirely in the surface, it is necessary (but not sufficient) that the Gaussian curvature $K(P)$ be nonpositive. Thus, you can conclude that if $K>0$ everywhere on $S$, there can be no lines contained in the surface. Ruled surfaces with $K=0$ everywhere are very special — they are called developable ruled surfaces and (locally) are either a plane, a (generalized) cylinder, a (generalized) cone, or the locus of tangent lines to a space curve. [See, for example, exercise 12 on p. 42 and exercise 10 on p. 65 of my differential geometry text.]

Your surface $S$ is given parametrically, but it's not hard to see that it satisfies the cartesian equation $2x+z=y^2$. A line through $P=(x_0,y_0,z_0)$ is given parametrically by $x=x_0+at$, $y=y_0+bt$, $z=z_0+ct$ for some scalars $a,b,c$. Your surface $S$ is given parametrically, but it's not hard to see that it is given by the cartesian equation $2x+z=y^2$. What happens when you substitute these into the equation of $S$? For the line to be wholly contained in $S$ the resulting equation must hold for all $t\in\Bbb R$.