In this question, it is asked how to prove that a doubly ruled surface by orthogonal lines is a plane.
I'm interested if this property is robust under perturbations of the angle the lines make; that is, suppose that we have a doubly ruled surface by lines which meet at angle $\theta$, where $\theta>0$ (so in the case of the question above, $\theta= \pi/2$). This $\theta$ is assumed to be fixed everywhere on the surface.
Question: with this assumption, can we conclude that the surface is a plane?
Any comment is appreciated!