The Magid-Ryan conjecture for \(4\)-dimensional affine spheres
type: Item
instance of: scholarly article
title: The Magid-Ryan conjecture for \(4\)-dimensional affine spheres
zbMATH Open document ID: 0956.53010
DOI: 10.1007/BF02940868
author: E. Bergen
author: E. Ramakers
author: Luc Vrancken
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2000-03-19T00:00:00Z
review text: A conjecture of \textit{M. Magid} and \textit{P. Ryan} [Trans. Am. Math. Soc. 330, 887-901 (1992; Zbl 0765.53009)] says that an affine hypersphere \(M^n\) in \(\mathbb{R}^{n+ 1}\) with constant sectional curvature \(a\) and with nonzero Pick invariant \(J\) is affinely equivalent with \((x^2_1\pm x^2_2)(x^2_3\pm x^2_4)\cdots(x^2_{2m- 1}\pm x^2_{2m})= 1\) (\(n+1= 2m\)) or with \((x^2_1\pm x^2_2)(x^2_3\pm x^2_4)\cdots (x^2_{2m- 1}\pm x^2_{2m}) x_{2m+ 1}= 1\) (\(n+1= 2m+1\)) while the curvature \(a\) vanishes. After this conjecture has been verified by Li, Penn, Simon, Magid, Yu, Ryan, Kriele and Vrancken in the cases of dimension \(n=2\) or \(n=3\) and in the cases of a positive definite or Lorentzian affine metric, the authors now show that it is also correct for dimension \(n=4\). Their proof consists of the choice of an appropriate local basis for the tangent bundle of \(M^4\).
Mathematics Subject Classification ID: 53A15
zbMATH DE Number: 1416700
zbMATH Keywords: constant affine sectional curvature
zbMATH Keywords: Magid-Ryan conjecture
zbMATH Keywords: affine hypersphere
zbMATH Keywords: Pick invariant
Wikidata QID: Q123181175
reviewed by: Kurt Leichtweiß
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02940868
OpenAlex ID: W2070436401
description: scientific article; zbMATH DE number 1416700