= Zero second fundamental form implies planar image
{title2=$II=0\Longrightarrow d\boldsymbol N=0$}
Differentiate the orthogonality of the unit normal to the two independent coordinate tangent vectors. If the <second fundamental form> vanishes, each derivative of the normal is orthogonal to both tangent vectors; differentiating its unit length also makes it orthogonal to the normal. Thus both normal derivatives vanish. On a connected parameter domain the normal is constant, and differentiating the position's scalar product with that constant normal proves the surface lies in a plane.
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