Preprint A257/2003
On real Kaehler Euclidean submanifolds with non--negative Ricci curvature

Fangyang Zheng | Florit, Luis A. | Hui, Wing San

**Keywords: **
Kaehler submanifolds | non-negative Ricci curvature | non-negative holomorphic curvature | relative nullity

We show that any Kaehler Euclidean submanifold with either non-negative Ricci curvature or non-negative holomorphic sectional curvature must has index of relative nullity
greater than or equal to 2n-2p, where 2n is the (real) dimension of the submanifold and p is its codimension. Moreover, if equality holds everywhere, then the submanifold must be a product of Euclidean hypersurfaces almost everywhere. We show also that the splitting is global provided the submanifold is complete. In particular, we conclude that the only Kaehler submanifolds with dimension 2n in Euclidean space of dimension 3n that have either positive Ricci curvature or positive holomorphic sectional curvature are the products of n orientable surfaces in $R^3$ with positive Gaussian curvature.