How does one characterize surfaces with constant nonzero Gaussian and mean curvature 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?
 A: In one codimension one can be convinced of this quite easily.  Consider the tracefree curvature tensor
$
A^o = A - \frac{1}{n}Hg,
$
where $g$ is the metric, $A$ the second fundamental form, and $H$ the mean curvature.  For surfaces, the norm squared of $A^o$ satisfies:
$
|A^o|^2 = (k_1 - k_2)^2,
$
where $k_1$ and $k_2$ are the principal curvatures.  On the other hand,
$
2|A^o|^2 = 2|A|^2 - H^2 = -2(H^2 - |A|^2) + H^2 = -4K + H^2 .
$
So if $K$ and $H$ are constant, then the norm of the tracefree curvature tensor is also constant.  Further, the symmetry of $(\nabla A)$ implies that every component of $A^o$ is constant, and so the principal curvatures are themselves constant functions.  If their difference is zero, then the principal curvatures are equal and it follows that the surface is a sphere or a plane.  In higher codimensions this is more complicated.  If their difference is not zero, this contradicts the compactness of the surface.
A: 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
