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!
Great question. Note that the reparametrization would have to leave the $s$-curves the same (so that the curvature functions match up). But this means we'd need to have the $E$s matching for the two surfaces, which we obviously don't.