On semi-Ryan complex submanifolds in an indefinite complex space form
type: Item
instance of: scholarly article
title: On semi-Ryan complex submanifolds in an indefinite complex space form
zbMATH Open document ID: 1002.53040
author: Young Suk Choi
author: Young Jin Suh
author: Jung-Hwan Kwon
published in: Rocky Mountain Journal of Mathematics
publication date: 2003-01-15T00:00:00Z
full work available at URL: CONT31-3.html
review text: Recently the present authors have given a complete classification of semisymmetric complex hypersurfaces, i.e. satisfying \(R(X,Y) \cdot R=0\), in a semidefinite complex space form \(M^m_1(c)\) [see the same journal, 31, 417-435 (2001)]. Now they give a complete classification of \(n\)-dimensional complete space-like complex submanifolds \(M\), satisfying \(R(X,Y) \cdot S=0\), where \(S\) is the Ricci tensor of \(M\) (such an \(M\) is called here semi-Ryan), in an \((n+p)\)-dimensional indefinite complex space form \(M^{n+p}_{0 +t}(c)\) of index \(2t\). It is shown that if the totally real holomorphic bisectional curvature of such an \(M\) is nonvanishing at all of its points, then \(M\) is Einstein. (The case of hypersurface has been investigated by \textit{A. Romero}, Math. Z. 192, 627-635 (1986; Zbl 0579.53019). In the real case the hypersurfaces with \(R(X,Y)\cdot S=0\) are classified by \textit{V. A. Mirzoyan}, Sb. Math. 191, 1323-1338 (2000; Zbl 1001.53006), who called them Ric-semiparallel). For an \(n\)-dimensional space-like semi-Ryan complex hypersurface \(M\) of the complex Minkowski space \(C^{n+1}_1\) the authors prove here that such an \(M\) is cylindrical.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 1744056
zbMATH Keywords: semi-definite complex submanifold
zbMATH Keywords: Ric-semi-symmetricity
zbMATH Keywords: semi-Ryan complex hypersurface
zbMATH Keywords: complex Minkowski space
reviewed by: Ülo Lumiste
MaRDI profile type: Publication
OpenAlex ID: W2009999241
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DOI: 10.1216/RMJM/1020171671
description: scientific article; zbMATH DE number 1744056