On real Kähler Euclidean submanifolds with non-negative Ricci curvature
type: Item
instance of: scholarly article
title: On real Kähler Euclidean submanifolds with non-negative Ricci curvature
zbMATH Open document ID: 1090.53010
DOI: 10.4171/JEMS/19
author: Luis A. Florit
author: Wing San Hui
author: Fangyang Zheng
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2005-03-21T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_abstract.php
review text: \textit{L. A. Florit} and \textit{F. Zheng} [Comment. Math. Helv. 74, No.1, 53--62 (1999; Zbl 0941.53014)] showed that, if \(f:M^{n}\rightarrow \mathbb R^{m+p}\), \(2p\leq m\), is an isometric immersion of an \(m\)-dimensional connected Riemannian manifold \(M^{m}\) with non-positive sectional curvature and negative Ricci curvature, into the Euclidean space \(\mathbb R^{m+p}\), then \(2p=m\) and \(f\) splits locally as a product of \(p\) surfaces in \(\mathbb R^{3}\). In this article, the authors study splitting results when the Riemannian manifold has a Kähler structure, and show: Let \(f:M^{2n}\rightarrow \mathbb R^{2n+p}\), \( p\leq n \), be an isometric immersion of a Kähler manifold with either non-negative Ricci curvature or non-negative\ holomorphic sectional curvature. Then the index of relative nullity \(\nu \) of \(f\) satisfies \(\nu \geq 2n-2p\). Moreover, if \(\nu =2n-2p\), then there is an open dense subset \( W\subset M^{2n}\) such that \(f\left| _{W}\right. \) splits locally as a product of \(p\) nowhere flat real Kähler Euclidean hypersurfaces with non-negative Ricci curvature.
reviewed by: Thomas Hasanis
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 2148274
zbMATH Keywords: Kähler manifold
zbMATH Keywords: isometric immersion
zbMATH Keywords: index of relative nullity
zbMATH Keywords: splitting
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2148274