A boundary uniqueness theorem for holomorphic functions of several complex variables

If $D\subset C^n$ is a region with a smooth boundary and $M\subset\partial D$ is a smooth manifold such that for some point $p\in M$ the complex linear hull of the tangent plane $T_p(M)$ coincides with $C^n$, then for each function $f\in A(D)$ the condition $f\mid_m=0$ implies that $f\equiv0$ in $D$.