# On real Kähler Euclidean submanifolds with non-negative Ricci curvature

Luis A. Florit; Wing San Hui; F. Zheng

Journal of the European Mathematical Society (2005)

- Volume: 007, Issue: 1, page 1-11
- ISSN: 1435-9855

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topFlorit, 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 -

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