Reducibility of complex submanifolds of the complex euclidean space

Abstract.

Let M be a simply connected complex submanifold of $\mathbb{C}^N$. We prove that M is irreducible, up a totally geodesic factor, if and only if the normal holonomy group acts irreducibly. This is an extrinsic analogue of the well-known De Rham decomposition theorem for a complex manifold. Our result is not valid in the real context, as it is shown by many counter-examples.