Global rigidity of compact classical Lie groups
type: Item
instance of: scholarly article
title: Global rigidity of compact classical Lie groups
zbMATH Open document ID: 0589.53057
author: Eiji Kaneda
published in: Hokkaido Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: Let G(n,F) denote the compact classical Lie group SO(n), U(n) or Sp(n) depending on \(F={\mathbb{R}}\), \({\mathbb{C}}\) or \({\mathbb{Q}}\), the field of quaternions. In this article the author deals with the problem of rigidity of G(n,F) imbedded isometrically in the space of \(n\times n\) matrices over F. We recall that an imbedding \(f: M\to {\mathbb{R}}^ m\), M a Riemannian manifold, is elliptic if i) the second fundamental form \(\alpha\) (x,y) generates the normal space at \(p\in M\), for all p; ii) for every normal direction \(\xi\), \(A_{\xi}\) has at least two non zero eigenvalues of the same sign \((<A_{\xi}X,Y>=<\alpha (x,y),\xi >)\). In Nagoya Math. J. 51, 137-160 (1973; Zbl 0265.53050), \textit{N. Tanaka} introduced the notion of elliptic imbedding and proved a rigidity theorem which may be stated as follows: Let f be an imbedding such that a) f is elliptic, b) it is infinitesimally rigid, c) M is compact. If \(f_ 1\) and \(f_ 2\) are two imbeddings of M in \({\mathbb{R}}^ m\) which lie near f with respect to the \(C^ 3\) topology and induce the same metric on M then there is a unique Euclidean transformation T of \({\mathbb{R}}^ m\) such that \(f_ 2=Tf_ 1.\) In the present paper it is proved that the inclusion map of SO(n), \(n\geq 5\), U(n), \(n\geq 3\), and Sp(n), \(n\geq i\) is elliptic and globally infinitesimally rigid hence rigid in the sense of Tanaka.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53A07
zbMATH DE Number: 3947124
zbMATH Keywords: Lie group
zbMATH Keywords: elliptic imbedding
zbMATH Keywords: rigidity theorem
zbMATH Keywords: infinitesimally rigid
reviewed by: Isabel Dotti Miatello
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1381757646
OpenAlex ID: W1975457785
Wikidata QID: Q115237728
DOI: 10.14492/HOKMJ/1381757646
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description: scientific article; zbMATH DE number 3947124