On locally symmetric affine hypersurfaces
type: Item
instance of: scholarly article
title: On locally symmetric affine hypersurfaces
zbMATH Open document ID: 0805.53008
DOI: 10.1007/BF01189575
author: Christine Scharlach
author: Luc Vrancken
published in: Archiv der Mathematik
publication date: 1994-11-13T00:00:00Z
review text: The authors investigate 3-dimensional, locally strongly convex affine hyperspheres \(M\) in \(\mathbb{R}^ 4\) for which the Levi-Civita connection of the affine metric is locally symmetric. Their main result is a complete classification of these hypersurfaces which gives a complete classification of the 3-dimensional, locally strongly convex affine homogeneous hyperspheres, too. They obtain three types of hypersurfaces, where two are isometric with spaces of constant sectional curvature and the third is isometric with the locally symmetric space \(\mathbb{R} \times H^ 2\). For the proof they use special adapted frames which were introduced in earlier works of the second author. Finally they give a 4- dimensional example of a locally strongly convex affine homogeneous hypersphere for which the Levi-Civita connection of the affine metric is not locally symmetric.
Mathematics Subject Classification ID: 53A15
zbMATH DE Number: 687033
zbMATH Keywords: affine hyperspheres
zbMATH Keywords: Levi-Civita connection
zbMATH Keywords: homogeneous hyperspheres
zbMATH Keywords: locally symmetric space
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 687033