We show that any real K\"ahler Euclidean submanifold \fk with either non-negative Ricci curvature or non-negative holomorphic sectional curvature has index of relative nullity greater than or equal to . Moreover, if equality holds everywhere, then the submanifold must be a product of Euclidean hypersurfaces almost everywhere, and the splitting is global provided that is complete. In particular, we conclude that the only real K\"ahler submanifolds in that have either positive Ricci curvature or positive holomorphic sectional curvature are precisely products of orientable surfaces in with positive Gaussian curvature. Further applications of our main result are also given.
Cite this article
Luis A. Florit, Wing San Hui, Fangyang Zheng, On real Kähler Euclidean submanifolds with non-negative Ricci curvature. J. Eur. Math. Soc. 7 (2005), no. 1, pp. 1–11DOI 10.4171/JEMS/19