mean curvature is not preserved under isometry

Question

I know that mean curvature is not intrinsic, so it seems that it is not preserved under isometry. What is the example that isometry does not preserve mean curvature?

Answer 1

Rolling up a plane into a cylinder can be given by the map $$\phi(x,y) = (\cos x,\sin x,y)$$ The derivatives are easily computed, $$\phi_{x} = (-\sin x,\cos x,0)$$ $$\phi_{y} = (0,0,1)$$ These are orthonormal with respect to the (induced) metric on the cylinder, so the parametrisation is an isometry. The plane has mean curvature $0$ while the cylinder has mean curvature $-1/2$