Minimal isometric immersions of spherical space forms in spheres
type: Item
instance of: scholarly article
title: Minimal isometric immersions of spherical space forms in spheres
zbMATH Open document ID: 0768.53028
DOI: 10.1007/BF02566512
author: Dennis DeTurck
author: Wolfgang Ziller
published in: Commentarii Mathematici Helvetici
publication date: 1993-01-11T00:00:00Z
full work available at URL: https://eudml.org/doc/140279
review text: A number of authors have studied minimal isometric immersions of Riemannian manifolds into round spheres, and in particular of round spheres into round spheres. As was observed by T. Takahashi, if \(\Phi: M \to S^ N(r) \subset \mathbb{R}^{N+1}\) is such a minimal immersion, then the components of \(\Phi\) must all be eigenfunctions of the Laplace operator on \(M\) corresponding to the same eigenvalue. The converse is correct. \textit{M. do Carmo} and \textit{N. Wallach} [Ann. Math., II. Ser. 93, 43-62 (1971; Zbl 0218.53062)] showed that in higher dimensions there are in general many minimal isometric immersions of \(S^ n(1)\) into \(S^ N(r)\), and they are parametrized by a compact convex body in a finite- dimensional vector space. In the present paper the following theorem is proved. Every homogeneous spherical space form admits a minimal isometric embedding into a standard sphere of appropriate radius. Half of the paper is devoted to the case of \(S^ 3\) separated from the rest because \(S^ 3\) is itself a group. The homogeneous three-dimensional space forms can be written as \(S^ 3/\Gamma\) where \(\Gamma\) is an arbitrary finite subgroup of \(S^ 3\). Thus the following groups are considered. The cyclic groups \(C_ d\). The binary dihedral groups \(D^*_ d\) and the groups \(T^*\), \(O^*\), \(I^*\). To summarize the results a table is presented. Some pages are devoted to higher-dimensional lens spaces and space forms of dimension \(4n-1\).
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 86166
zbMATH Keywords: isotropy-irreducible
zbMATH Keywords: isotropy group
zbMATH Keywords: Laplace operator
zbMATH Keywords: three-dimensional space forms
zbMATH Keywords: lens spaces
reviewed by: Yosio Mutō
MaRDI profile type: Publication
OpenAlex ID: W2033334413
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description: scientific article; zbMATH DE number 86166