The classification of 3-dimensional locally strongly convex homogeneous affine hypersurfaces
type: Item
instance of: scholarly article
title: The classification of 3-dimensional locally strongly convex homogeneous affine hypersurfaces
zbMATH Open document ID: 0795.53008
DOI: 10.1007/BF03026544
published in: Manuscripta Mathematica
publication date: 1994-02-07T00:00:00Z
full work available at URL: https://eudml.org/doc/155866
review text: The authors give a complete classification of the 3-dimensional homogeneous, locally strongly convex affine hypersurfaces of \(\mathbb{R}^ 4\), according to the number of different eigenvalues of their shape operator \(S\). So they prove that there are no 3-dimensional homogeneous, locally strongly convex affine hypersurfaces of \(\mathbb{R}^ 4\), whose shape operator \(S\) has three distinct eigenvalues. Next, considering that \(S\) has two distinct eigenvalues and being based on their previous papers they point out that a 3-dimensional homogeneous, locally strongly convex affine hypersurface \(M\) of \(\mathbb{R}^ 4\) is affine equivalent to the convex part of one of four explicit given hypersurfaces. Finally, if \(S\) is a multiple of the identity, \(M\) is a 3-dimensional locally strongly convex, locally homogeneous hypersphere in \(\mathbb{R}^ 4\) and then it is shown that either \(M\) is an open part of a locally strongly convex quadric or \(M\) is affine equivalent to an open part of one of the hypersurfaces \(xyzw=1\), \((y^ 2- z^ 2- w^ 2)^ 3x^ 2=1\).
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 492798
zbMATH Keywords: principal curvature
zbMATH Keywords: affine hypersurfaces
zbMATH Keywords: shape operator
zbMATH Keywords: quadric
author: Franki Dillen
author: Luc Vrancken
reviewed by: Despina Papadopoulou-Florou
MaRDI profile type: Publication
cites work: Affine hypersurfaces with parallel cubic forms
cites work: On the Pick invariant, the affine mean curvature and the Gauss curvature of affine surfaces
cites work: 3-dimensional affine hypersurfaces in ℝ4 with parallel cubic form
cites work: Calabi-type composition of affine spheres
cites work: Quasi-umbilical, locally strongly convex homogeneous affine hypersurfaces
cites work: A new model of unimodular-affinely homogeneous surfaces
cites work: Hyperbolic affine hyperspheres
cites work: Affine spheres with constant affine sectional curvature
description: scientific article; zbMATH DE number 492798