On a problem of Kuiper
type: Item
instance of: scholarly article
title: On a problem of Kuiper
zbMATH Open document ID: 1012.53053
DOI: 10.1007/BF02877013
published in: Science in China. Series A
publication date: 2002-08-15T00:00:00Z
review text: The purpose of the paper is to find a taut embedding of the complex quadrics into Euclidean spaces and thereby giving a positive answer to a question that \textit{N. H. Kuiper} asked on p.~115 in [Differential geometry, Proc. int. Chern. Symp., Berkeley 1979, 97-145 (1980; Zbl 0461.53033)]. The question was really superfluous since \textit{S. Kobayashi} had already proved in [Tohoku Math. J., II. Ser. 19, 63-70 (1967; Zbl 0149.18602)] that all compact homogeneous Kähler manifolds admit such taut embeddings; see also the paper [Bull. Soc. Math. Fr. 84, 251-281 (1956; Zbl 0073.40001)] of \textit{R. Bott} where essentially the same result is proved. To avoid further such misunderstanding, one should point out that all the other \` candidates\'\ on p.~115 in the paper of Kuiper can really be tautly embedded since they are special cases of so called \(R\)-spaces (also called generalized flag manifolds) that were proved to admit taut embeddings by \textit{R. Bott} and \textit{H. Samelson} in [Am. J. Math. 80, 964-1029 (1958; Zbl 0101.39702)] and independently by \textit{M. Takeuchi} and \textit{S. Kobayashi} in [J. Differ. Geom. 2, 203-215 (1968; Zbl 0165.24901)] One should note that the taut embedding of \(\text{SU}(n)\) is not the standard one into the matrix space \(M_n(\mathbb{C})\), but rather the one one gets by identifying it with the Stiefel manifold of \(n-1\) frames in \(\mathbb{C}^n\).
reviewed by: Gudlaugur Thorbergsson
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1782046
zbMATH Keywords: taut embeddings
zbMATH Keywords: complex quadrics
author: Kefeng Liu
MaRDI profile type: Publication
cites work: Q3911061
cites work: Q4308053
cites work: Minimum imbeddings of compact symmetric spaces of rank one
cites work: Q5510453
full work available at URL: https://doi.org/10.1007/bf02877013
OpenAlex ID: W2239169647
description: scientific article; zbMATH DE number 1782046