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I know that for any surface, the Gaussian curvature $K$ and mean curvature $H$ satisfy the inequality $H^2 \geq K$ , and the sphere is a surface where that inequality becomes an equation. Thus, the sphere has both constant Gaussian and mean curvature.

Are there other surfaces whose Gaussian and mean curvatures are constant and nonzero?

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    "Liebmann's theorem: The only regular (of class C²) closed surfaces in R³ with constant positive Gaussian curvature are spheres." http://en.wikipedia.org/wiki/Gaussian_curvature#Surfaces_of_constant_curvature2011-01-13

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