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I have the parametrization

$x(u,v)=(\cos u \sin v, \sin u \sin v , \cos v+\log (\tan {v/2}))$

with $0<v<\pi $ , $0<u<2\pi$.

From this parametrization, how can I compute (optimally) the gaussian curvature? I know for example that the pseudosphere is a revolution surface, then should exist a more easily way to calculate its curvature.

Thanks!

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  • $\begingroup$ Find the first and second fundamental forms. mathworld.wolfram.com/GaussianCurvature.html $\endgroup$ – David Peterson Nov 29 '14 at 2:11
  • $\begingroup$ For that parametrization is not very clever try to find directly the firts and the second fundamental form. $\endgroup$ – YTS Nov 29 '14 at 15:25
  • $\begingroup$ Throw darts at a dartboard then? $\endgroup$ – David Peterson Nov 29 '14 at 23:00
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Take $\partial_1=\frac{\partial x}{\partial u}$ and $\partial_2=\frac{\partial x}{\partial v}$ as the tangent frame. Then the normal will be $N=\frac{\partial_1\times\partial_2}{||\partial_1\times\partial_2||}$. Now the derivatives $D_{\partial_1}N=\frac{\partial N}{\partial u}$ and $D_{\partial_2}N=\frac{\partial N}{\partial v}$ are going to be tangent too, so you will get a base change $$D_{\partial_1}N=A\partial_1+B\partial_2$$ $$D_{\partial_2}N=C\partial_1+D\partial_2$$ for some scalars $A,B,C,D$. Then the determinant $AD-BC$ is the Gaussian curvature.

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  • $\begingroup$ This is not optimal, this is extremely long and tedius. $\endgroup$ – YTS Nov 29 '14 at 15:26
  • $\begingroup$ use an optimal parameterization, as in mathworld.wolfram.com/Pseudosphere.html $\endgroup$ – janmarqz Nov 29 '14 at 17:03

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