The differential of Gauss' map is zero if and only if $S$ is a plane

Question

Assume that the second fundamental form of a surface $S$ is identically equal to zero. It is well-known (and easy to be shown) that the surface is part of a plane! However, let me think geometrically to this situation. The Gauss' map $N$ is constant in this case, so that the span of $\sigma_u$ and $\sigma_v$ (here $\sigma$ denotes the parameterization of $S$) is always the same vector space. Now, from that, can I deduce that $\sigma_u$ and $\sigma_v$ are constant vectors?