Real submanifolds in complex spaces

Let (z ₁₁,..., z _1N ,..., z _m1,..., z _mN , w ₁₁,..., w _mm ) be the coordinates in \({\mathbb{C}^{mN + {m^2}}}\). In this note we prove the analogue of the Theorem of Moser in the case of the real-analytic submanifold M defined as follows

$$W = Z{\overline Z ^t} + O\left( 3 \right)$$

, where W = {w _ij }_1≤i,j≤m and Z = {z _ij }_{1≤i≤m, 1≤j≤N} . We prove that M is biholomorphically equivalent to the model \(W = Z{\overline Z ^t}\) if and only if is formally equivalent to it.