 Zero second fundamental form implies planar image

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.