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I'm preparing for an exam and I would like to know what are some examples of surfaces with constant Gaussian curvature such as surfaces with $k=0, \pm1$

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Have you computed many examples yourself? There are many common surfaces which have constant Gauss curvature, namely those you've listed in particular. In fact, there's a beautiful theorem that says that any compact surface with constant Gauss curvature $K > 0$ is part of a sphere of radius $\frac{1}{\sqrt{K}}$. I'll post a more detailed answer below once you update. – Alex Wertheim May 1 '13 at 3:53
May be you meant "locally isometric to", but not "part of ", which implies rigidity. – Narasimham Aug 29 '14 at 19:22
up vote 1 down vote accepted

HINT: try to compute the Gaussian curvature for the following objects (embedded) in $\mathbb{R}^3$ using parametrization:

  • Plane, cylinder, cone(away from the vertex)

  • Sphere

  • Tractrix, tractricoid

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