Why is the tube rigid?

by Marek Kordos

Take a surface and at some point cut it with all the planes that contain the straight line orthogonal to the surface at that point. Among the curves so generated there is one with the greatest curvature and one with the least. Even more: if the curvatures of those curves are different, the curves with extreme curvatures are orthogonal. Gauss has proved (in what is known as theorema egregium, the "wonderful theorem") that when a surface is being bent so that no curve on the surface changes its length, the product of extreme curvatures remains constant.

Thus if a surface (e.g. a tube) is made of some flat material, then at each point one of its extreme curvatures must be equal to 0, to preserve the constant value of the product, equal to 0 on the plane (where the curvatures are equal to 0). This implies that in order to curve a tube (i.e. to bend the generator of the cylinder) you must first stretch the material it is made of.