A local and global splitting result for real Kähler Euclidean submanifolds

Abstract.

We show that if a real Kähler Euclidean submanifold is as far as possible of being minimal, then it should split locally as a product of hypersurfaces almost everywhere, possibly in lower codimension. In addition, if the manifold is complete, simply connected and has constant nullity, it should split globally as a product of surfaces in $\mathbb{R}^3$ and an Euclidean factor. Several applications are also given.