On real Kähler Euclidean submanifolds with non-negative Ricci curvature

Florit, Luis A., Hui, Wing San, and Zheng, F.. "On real Kähler Euclidean submanifolds with non-negative Ricci curvature." Journal of the European Mathematical Society 007.1 (2005): 1-11. <http://eudml.org/doc/277302>.

@article{Florit2005,
abstract = {We show that any real Kähler Euclidean submanifold $f:M^\{2n\}\rightarrow \mathbb \{R\}^\{2n+p\}$ 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 products of n orientable surfaces in $\mathbb \{R\}^3$ with positive Gaussian curvature. Further applications of our main result are also given.},
author = {Florit, Luis A., Hui, Wing San, Zheng, F.},
journal = {Journal of the European Mathematical Society},
keywords = {Kähler submanifolds; Ricci curvature; holomorphic curvature; splitting; Kähler manifold; isometric immersion; index of relative nullity; splitting},
language = {eng},
number = {1},
pages = {1-11},
publisher = {European Mathematical Society Publishing House},
title = {On real Kähler Euclidean submanifolds with non-negative Ricci curvature},
url = {http://eudml.org/doc/277302},
volume = {007},
year = {2005},
}

TY - JOUR
AU - Florit, Luis A.
AU - Hui, Wing San
AU - Zheng, F.
TI - On real Kähler Euclidean submanifolds with non-negative Ricci curvature
JO - Journal of the European Mathematical Society
PY - 2005
PB - European Mathematical Society Publishing House
VL - 007
IS - 1
SP - 1
EP - 11
AB - We show that any real Kähler Euclidean submanifold $f:M^{2n}\rightarrow \mathbb {R}^{2n+p}$ 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 products of n orientable surfaces in $\mathbb {R}^3$ with positive Gaussian curvature. Further applications of our main result are also given.
LA - eng
KW - Kähler submanifolds; Ricci curvature; holomorphic curvature; splitting; Kähler manifold; isometric immersion; index of relative nullity; splitting
UR - http://eudml.org/doc/277302
ER -