A Helmholtz-Lie type characterization of ellipsoids. I
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.
scientific article; zbMATH DE number 764241
scientific article