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

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.

• Why surfaces : already plane curve, two curves are locally isometric, but they have the same curvature iff they are the same up to an isometry of the plane. A line (0 curvature) and a circle (of radious r, curvaturee r^2) are locally isometric ! – Thomas Apr 20 '18 at 4:27
• I am a newly joined Research Scholar. My field of research is Curve and Surface. I want to do something on Isometry, First and Second Fundamental form, Gaussian Curvature. If you suggest some books or article I will be very helpful. Or plz give me some suggestion how to go ahed with this topic. – Pinaki Ranjan Ghosh Apr 23 '18 at 7:59
• Manfredo Do Carmo, Differential geometry of curves and surfaces. – Thomas Apr 23 '18 at 14:13
• Thank you very much Sir. – Pinaki Ranjan Ghosh Apr 24 '18 at 6:57
• @Sebastiano: Please don't edit three-year-old posts unless you're making a substantial mathematical improvement. It brings them up to the front of the line and for absolutely no good reason. – Ted Shifrin Jan 15 at 23:29

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: $$L==-1 \neq 0$$.