How can you show that two surfaces are not isometries of each other, but have the same Gaussian Curvature. For example, I see that:
the helicoid given by X = (ucosv, usinv, v)
the surface Y = (ucosv, usinv, ln(u))
have the same Gaussian curvature. I computed the first and second fundamental forms and noticed that
K = -1/(1+u^2)^2
for both the helicoid (X) and the other surface (Y). I know that they are not isometries, but I am not sure how to show that there is no local reparametrization of X that has the first fundamental form equal to Y's first fundamental form. Any hints would be greatly appreciated! Thank you!