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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.

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  • $\begingroup$ 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 ! $\endgroup$ – Thomas Apr 20 '18 at 4:27
  • $\begingroup$ 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. $\endgroup$ – Pinaki Ranjan Ghosh Apr 23 '18 at 7:59
  • $\begingroup$ Manfredo Do Carmo, Differential geometry of curves and surfaces. $\endgroup$ – Thomas Apr 23 '18 at 14:13
  • $\begingroup$ Thank you very much Sir. $\endgroup$ – Pinaki Ranjan Ghosh Apr 24 '18 at 6:57
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    $\begingroup$ @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. $\endgroup$ – Ted Shifrin Jan 15 at 23:29
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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: $L=<U,\mathbb{x}_{uu}>=-1 \neq 0$.

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