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We show that any real Kähler Euclidean submanifold $f:M^{2n}\to {ℝ}^{2n+p}$ 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 ${ℝ}^{3n}$ that have either positive Ricci curvature or...