A Helmholtz-Lie type characterization of ellipsoids. I
type: Item
instance of: scholarly article
title: A Helmholtz-Lie type characterization of ellipsoids. I
zbMATH Open document ID: 0824.52006
DOI: 10.1007/BF02574059
author: Peter M. Gruber
published in: Discrete \& Computational Geometry
publication date: 1995-07-02T00:00:00Z
full work available at URL: https://eudml.org/doc/131377
review text: Results of Helmholtz-Lie type are characterizations of special quadrics or classical spaces and geometries by the property that there are local or global self-mappings of the surface or space which preserve the local geometric structure. The aim of this article is to prove the following Theorem: ``A closed convex surface \(S\) in \(E^ d\) with \(d\) odd, is an ellipsoid if and only if it has the following property: for any pair of points \(x\), \(y\) in \(S\) there is an affine transformation which maps \(x\) onto \(y\) and a suitable neighborhood of \(x\) in \(S\) onto a neighborhood of \(y\) in \(S\).'' The proof makes use of the differentiability results of Alexandrov for convex surfaces, of a characterization of ellipsoids due to Burton by the property that their planar sections are centrally symmetric, and of a topological result of Mani on continuous fields of congruent planar convex curves tangent to the unit sphere \(S^{d-1}\) (\(d\) odd). Besides, Baire's category theorem and a result on invariant subspaces of symmetric matrices are used. Remark: The case ``\(d\) even'' will be dealt with in the second part of this article (to appear) under the additional assumption that the affinities are volume-preserving.
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 51N10
Mathematics Subject Classification ID: 53C45
zbMATH DE Number: 764241
zbMATH Keywords: Euclidean space
zbMATH Keywords: Helmholtz-Lie type characterization
zbMATH Keywords: closed convex surface
zbMATH Keywords: ellipsoid
zbMATH Keywords: affine transformation
MaRDI profile type: Publication
cites work: Sur une propriété caractéristique de l'ellipsoїde
cites work: Seules les quadriques admettent des caustiques
cites work: On the sum of a zonotope and an ellipsoid
cites work: Congruent sections of a convex body
cites work: Q5538720
cites work: Über das Löwnersche Ellipsoid und sein Analogon unter den einem Eikörper einbeschriebenen Ellipsoiden
cites work: Neuere Fassungen des Riemann-Helmholtz-Lieschen Raumproblems
cites work: Q3359644
cites work: Über kennzeichnende Eigenschaften von euklidischen Räumen und Ellipsoiden. I.
cites work: Q4089429
cites work: Q5514233
cites work: Fields of planar bodies tangent to spheres
cites work: Q3667512
cites work: Q3862690
cites work: Q5287551
cites work: Error and Perturbation Bounds for Subspaces Associated with Certain Eigenvalue Problems
cites work: Kennzeichnende Eigenschaften der Kugel als Folgerung eines Brouwerschen Fixpunktsatzes
OpenAlex ID: W1989673377
description: scientific article; zbMATH DE number 764241