On real Kähler Euclidean submanifolds with non-negative Ricci curvature

Abstract

We show that any real Kähler Euclidean submanifold $f:M^{2n}\to {\mathbb{R}}^{2n+p}$ with either non-negative Ricci curvature or non-negative holomorphic sectional curvature has index of relative nullity greater than or equal to $2n-2p$. Moreover, if equality holds everywhere, then the submanifold must be a product of Euclidean hypersurfaces almost everywhere, and the splitting is global provided that $M^{2n}$ is complete. In particular, we conclude that the only real Kähler submanifolds $M^{2n}$ in ${\mathbb{R}}^{3n}$ that have either positive Ricci curvature or positive holomorphic sectional curvature are precisely products of $n$ orientable surfaces in ${\mathbb{R}}^3$ with positive Gaussian curvature. Further applications of our main result are also given.