Classification of flat indefinite equicentroaffinely homogeneous surfaces in \(\mathbf R^ 4\)
type: Item
instance of: scholarly article
title: Classification of flat indefinite equicentroaffinely homogeneous surfaces in \(\mathbf R^ 4\)
zbMATH Open document ID: 0860.53009
DOI: 10.1007/BF03322225
published in: Results in Mathematics
publication date: 1997-04-20T00:00:00Z
review text: Let \(x: M\to \mathbb{R}^4\) be an immersion of a surface. For any local basis \(\sigma=\{E_1,E_2\}\) one can locally define a symmetric 2-form \(G^\sigma=[E_1(x),E_2(x),x,d^2x] =\Sigma G^\sigma_{ij} \theta_i \theta_j\), where \(\{\theta_1,\theta_2\}\) is the dual basis for \(\sigma\). \(x\) is called nondegenerate if \(G^\sigma\) is a nondegenerate 2-form. It is known that for any nondegenerate surface \(x\) one gets a globally defined equi-centroaffinely invariant metric \(G:=|\text{det}G^\sigma_{ij}|^{1/4}G^\sigma\). The author uses \(G\) to define a complete system of equi-centroaffine invariants \(\{\Lambda,\Phi,\Psi,\Theta\}\) (where \(\Phi\) and \(\Lambda\) are 3-forms, \(\Psi\) and \(\Theta\) are 2-forms), which determines the surface \(x\) up to equi-centroaffine transformations in \(\mathbb{R}^4\). Using this invariant system, the author is able to determine all the nondegenerate flat equi-centroaffine homogeneous surfaces in \(\mathbb{R}^4\) which are not contained in any hyperplane of \(\mathbb{R}^4\). There are 16 types of such homogeneous surfaces. The author also gives examples of equi-centroaffine homogeneous surfaces which are orbits of two different centroaffine subgroups.
reviewed by: Changping Wang
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 920812
zbMATH Keywords: equicentroaffine geometry
zbMATH Keywords: equicentroaffine invariants
zbMATH Keywords: Euclidean four-space
zbMATH Keywords: homogeneous surfaces
author: Huili Liu
MaRDI profile type: Publication
cites work: Q5725219
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full work available at URL: https://doi.org/10.1007/bf03322225
OpenAlex ID: W2038885158
description: scientific article; zbMATH DE number 920812