We consider the class of surfaces that have zero Gaussian curvature. By putting the metric into the form Edu^2 + Gdv^2, we can show that either L or N in the second fundamental form must be zero (M is automatically zero by this construction). Once one of the two coefficients in the second fundamental form is zero, we can show that one of the two ruling of the surface are straight lines. This proves that the surface is a ruled surface and must be a cylinder, cone or a tangent developable.
#mikethemathematician, #mikedabkowski, #profdabkowski, #differentialgeometry
