Real submanifolds in complex spaces

Abstract

Let (z11,..., z1N,..., zm1,..., zmN, w11,..., wmm) 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 = {wij}1≤i,j≤m and Z = {zij}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.