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I am trying to see how to compute the First Fundamental Form of this set: $\{z_1^2+z_2^2=1,(z_1,z_2) \in \mathbb C^2 \}$ . First I have seen in this post is homeomorphic to the sphere without two points. But then the parametrization I have seen depends of 1 complex number.

Is it enought to compute the first fundamental form in the sphere without two points and multiplied by something to preserve the First Fundamental Form?

Thanks in advance.

energy
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  • It would help if you added your definition for the first fundamental form. – Amitai Yuval Feb 02 '19 at 14:59
  • I only know this definition https://en.wikipedia.org/wiki/First_fundamental_form – energy Feb 02 '19 at 15:28
  • The Wiki page you quote concerns surfaces in three-dimensional Euclidean space. Your post seems to ask about complex curves in two-dimensional complex space. – Amitai Yuval Feb 02 '19 at 16:20
  • But it is homeomorphic to a sphere without two points so you could define the first fundamental form right? – energy Feb 02 '19 at 18:20

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