On the nullity index of isometric immersions of Kähler manifolds
type: Item
instance of: scholarly article
title: On the nullity index of isometric immersions of Kähler manifolds
zbMATH Open document ID: 1039.53067
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2003-10-19T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108547
review text: Let \(\varphi:M\to N\) be an isometric immersion of a Kähler manifold with complex dimension \(m\) into an ambient space \(N\). The map \(\varphi\) is said to be \textit{pluriminimal} if, for any holomorphic curve \(c\) immersed in \(M\), the curve \(\varphi\circ c\) is minimal. When \(N\) is flat, \textit{M. Dajczer} and \textit{L. Rodríguez} [Duke Math. J. 53, 211--220 (1986; Zbl 0599.53005)] showed that the pluriminimal immersions are exactly the minimal ones. Set \(\nu(x)\) (resp. \(\nu^{(1,1)}(x)\)) the index of relative nullity of the second fundamental form \(\alpha\) of \(\varphi\) (resp. of the \((1, 1)\) component of \(\alpha\)) at \(x\in M\). In case \(N=\mathbb{R}^{2n +p}\), \(n\geq 2\), M. Dajczer and L. Rodríguez classified the minimal immersions when \(\nu\) is greater than \(2n-4\) everywhere. In this article \(N\) is a conformally flat manifold having \(\nu(x_0)>0\) or \(\nu^{(1,1)}(x_0)>0\) at some point \(x_0\in M\). The authors show, under certain conditions, that \(m=1\), i.e. \(M\) is a Riemannian surface.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53B20
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1993729
zbMATH Keywords: nullity index
zbMATH Keywords: isometric immersion
zbMATH Keywords: conformally flat manifold
author: Maria João Ferreira
author: R. Tribuzy
reviewed by: Marian Ioan Munteanu
MaRDI profile type: Publication
cites work: Q3274628
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cites work: Q4011943
description: scientific article; zbMATH DE number 1993729