Let $\Sigma$ be a $C''$ surface defined on an open connected set D in the UV plane. Suppose $d^2\Sigma=0$ in $D$,prove that $\Sigma$ is a plane.
I know $\Sigma$ has the form
$\Sigma(u,v)=(x_1(u,v),x_2(u,v),x_3(u,v))$
Therefore,
$\displaystyle{\frac{\partial^2 x_i}{\partial u^2}=\frac{\partial^2 x_i}{\partial v^2}=\frac{\partial^2 x_i}{\partial u\partial v}=\frac{\partial^2 x_i}{\partial v\partial u}=0}$, for $i=1,2,3$
From that I can imply that $\Sigma$ only depends in a linear way from each variable? How is the open connected hypothesis use?