Indefinite Einstein hypersurfaces with nilpotent shape operators
type: Item
instance of: scholarly article
title: Indefinite Einstein hypersurfaces with nilpotent shape operators
zbMATH Open document ID: 0541.53047
author: M. Magid
published in: Hokkaido Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: Let \(f:M_ n^{2n}\to N^{2n+1}(c)\) be an isometric immersion of an Einstein hypersurface in an indefinite space form \(N^{(2n+1)}(c)\) (c denoting a non zero constant), and let A be the shape operator associated with f. In a previous paper [Proc. Am. Math. Soc. 84, 237-242 (1982; Zbl 0485.53023)], the author proved that if A is not diagonalizable, then there are two possibilities: either \(A^ 2=0\) or \(A^ 2=-b^ 2I\) (b denoting a non zero constant). In the present paper those \(M_ n^{2n}\) with \(A^ 2=0\) and maximal rank of A are classified. We shall quote here the following two interesting results: (i) If \(A^ 2=0\) and rank A\(=n\), then Ker(A) is an integrable totally isotropic (i.e. Ker(A)\(\subset orth.Ker(A))\) and parallel n-dimensional distribution on \(M_ n^{2n} (M_ n^{2n}\) is of signature (n,n)). (ii) Let \(f:M_ n^{2n}\to R_ n^{2n+1}\) be an isometric immersion with rank A\(=n\). Then Ker(A) is an integrable, totally isotropic, parallel distribution on \(M_ n^{2n}\), iff \(A^ 2=0\).
reviewed by: R. Rosca
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 3859935
zbMATH Keywords: nilpotent shape operators
zbMATH Keywords: B-scroll
zbMATH Keywords: isometric immersion
zbMATH Keywords: Einstein hypersurface
zbMATH Keywords: indefinite space form
zbMATH Keywords: parallel distribution
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1381757717
OpenAlex ID: W2044292342
DOI: 10.14492/HOKMJ/1381757717
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description: scientific article; zbMATH DE number 3859935