TITLE: How does one characterize surfaces with constant nonzero Gaussian and mean curvature
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: The only surfaces in Euclidean space with $K$ and $H$ both constant are: planes, spheres and right circular cylinders. It appears as an exercise in the Struik's book and a proof in "Curves and Surfaces", Montiel-Ros, Graduate Studies in Mathematics, vol. 69. AMS, 2009