A characterization of submanifolds of extrinsic symmetric spaces
type: Item
instance of: scholarly article
title: A characterization of submanifolds of extrinsic symmetric spaces
zbMATH Open document ID: 1191.53034
author: J.-H. Eschenburg
author: R. Tribuzy
published in: Differential Geometry and its Applications
publication date: 2010-05-28T00:00:00Z
review text: A submanifold \(S\) of a Euclidean space is said to be a full extrinsically symmetric space if \(S\) is compact and the second fundamental form \(\beta^S\) of \(S\) is parallel and onto. Let \(M\) be a submanifold of \({\mathbb R}^n\) and \(S\) a full extrinsically symmetric space in \({\mathbb R}^n\). Assume that there exist a subbundle \(E \subset M \times {\mathbb R}^n\) with \(TM \subset E\) and a parallel bundle homomorphism \(\beta : E \otimes E \to E^\perp\) satisfying \(\beta(v,a) = (\partial_va)^{E^\perp}\) for all \(v \in TM\) and \(a \in E\). Assume furthermore that \(\beta\) is congruent to \(\beta^S\) at some point. The author proves that \(M\) is contained in \(S\) up to a rigid motion of \({\mathbb R}^n\). This generalizes the well-known result for the special case when \(S = S^{n-1} \subset {\mathbb R}^n\).
reviewed by: Jürgen Berndt
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53C05
zbMATH DE Number: 5713585
zbMATH Keywords: vector subbundles
zbMATH Keywords: connections
zbMATH Keywords: Gauss equations
MaRDI profile type: Publication
Wikidata QID: Q115357123
full work available at URL: j.difgeo.2010.01.002
OpenAlex ID: W1979364316
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cites work: Symmetric submanifolds of Euclidean space
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DOI: 10.1016/J.DIFGEO.2010.01.002
description: scientific article; zbMATH DE number 5713585