TITLE: Two surfaces are not isometries of each other, but have the same Gaussian Curvature
QUESTION [7 upvotes]: 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!

REPLY [3 votes]: 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.