Gaussian curvature, mean curvature I'm looking for some examples of surfaces, where the Gaussian curvature $K$ is zero, but the mean curvature $H$ is not. And vice versa: Are there any examples of surfaces where $H = 0$, but $K$ is not zero? 
And: Why does the Weingarten map vanishes at all points of the (connected smooth) surface, if $K = H = 0$? 
Thank you for every contribution! 

The Gaussian and mean curvatures are defined as $K=k_1k_2$, $\;H={1\over 2}(k_1+k_2)$ respectively, where $k_1$ and $k_2$ are the principal curvatures.
Clearly, $K$ vanishes when any of the principal curvatures is zero (or both) and $H$ does so when the principal curvatures are the opposite of each other. If both principal curvatures are zero, then $K=H=0$ and the Weingarten map vanishes since the eigenvalues of it are precisely $k_1$ and $k_2$.
There are many examples for you. Here go two:
A cylinder is a surface in $\mathbb{R}^3$ with zero Gaussian curvature but mean curvature ${1\over 2r}$, where $r$ is the radius.
A catenoid is also a surface in $\mathbb{R}^3$ with Gaussian curvature given by
$$-{sech^4({v\over r})\over r^2},$$
where $r$ is the radius and $v$ is one of the parameters of the usual parametrization, and the mean curvature is zero.
