Flat hyperbolic Centro-affine Tchebychev hypersurfaces of \(\mathbb{R}^4\)
type: Item
instance of: scholarly article
title: Flat hyperbolic Centro-affine Tchebychev hypersurfaces of \(\mathbb{R}^4\)
zbMATH Open document ID: 1480.53006
author: Sébastien Lalléchère
author: Lucius Ramifidisoa
author: Blaise Ravelo
published in: Results in Mathematics
publication date: 2021-07-07T00:00:00Z
review text: Let \(M\) be a hyperbolic centro-affine hypersurface in \(\mathbb{R}^4\), and let \(X, Y, Z\) be vector fields in \(T_pM\). Assume that \(M\) is flat, in other words, that the curvature tensor \(R\) associated with the Levi-Civita connection of the centro-affine metric satisfies \(R(X, Y )Z = 0\). Assume also that \(M\) is a Tchebychev surface in the sense of \textit{S. L. Samelson} [Arch. Ration. Mech. Anal. 114, No. 3, 237--254 (1991; Zbl 0727.53041)]. That is, that the Tchebychev vector field \(T^{\#}\) satisfies \(\hat{\nabla}_X T^{\#} = \alpha X\) where \(\alpha\) is a differentiable function on \(M\). The main theorem in this article is an explicit classification of such hypersurfaces up to centro-affine equivalence. The classification consists of three explicit canonical families which coincide with the lists found in the related problems of \textit{H. Trabelsi} [Result. Math. 51, No. 1--2, 127--140 (2007; Zbl 1147.53010)] and \textit{A. Li} and \textit{C. Wang} [Result. Math. 20, No. 3--4, 660--681 (1991; Zbl 0752.53010)].
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 53B05
zbMATH DE Number: 7369083
zbMATH Keywords: curvature tensor
zbMATH Keywords: hyperbolic hypersurface
zbMATH Keywords: centro-affine hypersurface
zbMATH Keywords: Tchebychev
zbMATH Keywords: Levi-Civita connection
Wikidata QID: Q115609468
reviewed by: Graham M. Reeve
MaRDI profile type: Publication
full work available at URL: s00025-021-01363-z
OpenAlex ID: W3138573765
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DOI: 10.1007/S00025-021-01363-Z
description: scientific article; zbMATH DE number 7369083