k-symmetric submanifolds of \(R^ N\)
type: Item
instance of: scholarly article
title: k-symmetric submanifolds of \(R^ N\)
zbMATH Open document ID: 0536.53052
DOI: 10.1007/BF01456188
author: Cristián U. Sánchez
published in: Mathematische Annalen
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/182960
review text: This paper studies submanifolds \(M^ n\) of \(R^ N\) which are k- symmetric and such that the corresponding isometry \(\Theta_ p\) at each point extends to an isometry \(\Omega_ p: R^ N\to R^ N\) which is the identity on the normal space to \(M^ n\) at the point. The case \(k=2\) was considered by \textit{D. Ferus} [Math. Ann. 247, 81-93 (1980; Zbl 0446.53041)]. He obtains a complete classification by using the fact that these submanifolds have parallel second fundamental form. But for \(k\neq 2\) these submanifolds do not have parallel second fundamental form and then quite different methods are required. The purpose of this paper is to study the case k odd. It contains a complete description of the compact k-symmetric spaces which admit imbeddings of this type as well as a description of the imbeddings. The methods are those of the theory of representations of Lie groups and algebras. The paper also contains results about the total absolute curvature of these submanifolds.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A07
zbMATH DE Number: 3851963
zbMATH Keywords: classification
zbMATH Keywords: compact k-symmetric spaces
zbMATH Keywords: representations of Lie groups
zbMATH Keywords: total absolute curvature
MaRDI profile type: Publication
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OpenAlex ID: W1982193917
description: scientific article; zbMATH DE number 3851963