Is there any example of two surface having same first fubndamental form are whose second fundamental form are different.

Question

I know that two surfaces are isometric if and only if they have same first fundamental form. I can not think of any examples of two surfaces in $R^3$ which have the same first fundamental form but different second fundamental form. Is there an example of two surfaces with this property?

Thank you.

Answer 1

Compare $\mathbb{R}^2$ and the cylinder $\mathbb{x}(u,v)=(\cos u, \sin u, v)$. They have the same first fundamental form given by: $E=G=1,F=0$. But the second fundamental form of a cylinder is: $L=<U,\mathbb{x}_{uu}>=-1 \neq 0$.