If the derivative of the Gauss map is zero in every point in the image of a given local chart, can I conclude that the normal vector is constant and such image is contained in a plane?
Edit: The question in my homework is the following: if the second fundamental form of a surface patch for a surface is idenportically zero, then the image of the surface patch is contained in a plane.
Given the relation between the second fundamental form and the derivative of the Gauss map, I proved that the derivative of the Gauss map is zero in every point. Can I conclude?