On normally flat Einstein submanifolds
type: Item
instance of: scholarly article
title: On normally flat Einstein submanifolds
zbMATH Open document ID: 0882.53018
DOI: 10.1155/S0161171297000677
author: Leopold Verstraelen
author: Georges Zafindratafa
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1998-03-05T00:00:00Z
full work available at URL: https://eudml.org/doc/47817
review text: Let \(M\) be a Riemannian 4-manifold isometrically immersed with flat normal connection in Euclidean 6-space \(E^6\). In terms of the Weingarten operators \(A_{ N_1}\), \(A_{N_2}\), which can be diagonalized simultaneously, as is well-known, a classification is given for the case of Einstein manifolds \(M\). An example is given showing that the scalar curvature of such an immersed \(M\) can be any given real number (if the codimension is 1, it must be positive!). This classification is then specified for the case of Ricci-flat \(M\). It is shown that an \(n\)-dimensional Riemannian manifold \(M\), isometrically immersed with flat normal connection in \(E^{n+N}\), is flat if and only if it is cylindrical, i.e., if there exists an orthonormal frame field with directions \(\xi\) such that every \(A_\xi\) admits a zero eigenvalue with multiplicity of \(n-1\) or \(n\). The relation with Cartan's conjecture (that there exist flat non-cylindrical submanifolds of dimension \(>3\) in Euclidean spaces) was previously discussed. [See also: \textit{Ü. Lumiste}, Sov. Math. 31, 23-35 (1987; Zbl 0616.53006)].
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 1054255
zbMATH Keywords: Einstein submanifold
zbMATH Keywords: Ricci flat submanifold
zbMATH Keywords: flat normal connection
reviewed by: Ülo Lumiste
MaRDI profile type: Publication
OpenAlex ID: W2124215350
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description: scientific article; zbMATH DE number 1054255