On real Kähler Euclidean submanifolds with non-negative Ricci curvature

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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 \R^{3n} that have either positive Ricci curvature or positive holomorphic sectional curvature are precisely products of n orientable surfaces in \R^3 with positive Gaussian curvature. Further applications of our main result are also given.