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I'd like help with the following question:

Prove that all revolution surfaces $(\phi(v) \cos u ,\phi(v) \sin u,\psi(v)) $ of constant Gaussian curvature $k = -1$ is one of the following types:

  1. $\phi(v)=C\cosh v$ and $\psi(v)=\int_0^v \sqrt{1-C^2\sinh^2v} dv$

  2. $\phi(v)=C\sinh v$ and $\psi(v)=\int_0^v \sqrt{1-C^2\cosh^2v} dv$

  3. $\phi(v)=e^v$ and $\psi(v)=\int_0^v \sqrt{1-e^{2v} dv}$

Suppose $(\phi')^2+(\psi')^2=1$ and you know that $\phi''+k\phi= 0$

thanks

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