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Is there a surface with first $\Phi_1=(\text{d}u)^2+(\text{d}v)^2$ and second $\Phi_2=(\text{d}u)^2-(\text{d}v)^2$ fundamental form?

I can give examples of surface with the first (x)or the other form. But how can I combine these together?

Thanks.

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  • $\begingroup$ Think about what these equations imply about the curvature - in particular, the Gauss equation should be quite simple to compute and quite disturbing to behold. $\endgroup$ Jun 20, 2017 at 12:58

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No there isn't. You can prove this by contradiction. You have $e=1,g=-1,f=0,E=1,F=0,G=1$. Hence the guassian curvature of your surface is $$\frac{eg-f^2}{EG-F^2}= -1$$

But you know that gaussian curvature is an intrinsic property of a surface, which means that it only depends on the first fundamental form. Now, you probably don't remember that formula for curvature in terms of $E,F,G$ and their derivatives only. So, for your own good look at it and calculate the curvature with it. I think it will be $0$.(haven't checked, just guessing value from experience :) )

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  • $\begingroup$ I looked for a formula. But I could not find it in my textbook. (Of course, I might overlooked that.) Which relation do you mean? $\endgroup$
    – byk7
    Jun 20, 2017 at 16:12
  • $\begingroup$ @byk7: there are a few formulas here that would do the job. $\endgroup$ Jun 22, 2017 at 3:57

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